I. A power series is a polynomial with infinitely many terms of the form Σn=0∞anxn.
II. The radius of convergence R determines the values of x where a power series converges absolutely (for |x|<R), diverges (for |x|>R), or may converge or diverge (for |x|=R).
III. Tests like the ratio test and root test can be used to calculate the radius of convergence R.
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In this paper, we introduce a new approach on the convex orderings and integral inequalities of the convex orderings of the triangular fuzzy random variables. Based on these orderings, some theorems and integral inequalities are established.
A New Approach on the Log - Convex Orderings and Integral inequalities of the...inventionjournals
In this paper, we introduce a new approach on the convex orderings and integral inequalities of the convex orderings of the triangular fuzzy random variables. Based on these orderings, some theorems and integral inequalities are established.
In this slide i am trying my best to describe about the power series. If you face any problem or anything that you can't understand please contact me on facebook:https://www.facebook.com/asadujjaman.asad.79
In this slide i am trying my best to describe about the power series. If you face any problem or anything that you can't understand please contact me on facebook:https://www.facebook.com/asadujjaman.asad.79
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
1. A power series P(x) is a "polynomial" in x of infinitely
many terms.
Power Series
2. A power series P(x) is a "polynomial" in x of infinitely
many terms. That is,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….
Power Series
P(x) =
3. A power series P(x) is a "polynomial" in x of infinitely
many terms. That is,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….
Note that P(0) = a0.
Power Series
P(x) =
4. A power series P(x) is a "polynomial" in x of infinitely
many terms. That is,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….
The basic question for a power series is to determine
the value(s) of x where it converges or diverges.
Note that P(0) = a0.
Power Series
P(x) =
5. A power series P(x) is a "polynomial" in x of infinitely
many terms. That is,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….
Example: A. Let P(x) = 1 + x + x2 + x3 + x4 …
The basic question for a power series is to determine
the value(s) of x where it converges or diverges.
Note that P(0) = a0.
Power Series
P(x) =
6. A power series P(x) is a "polynomial" in x of infinitely
many terms. That is,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….
Example: A. Let P(x) = 1 + x + x2 + x3 + x4 …
From the theorem of geometric series, given x = r,
I. P(r) = 1 + r + r2 + r3 + r4 + … converges if | r | < 1.
The basic question for a power series is to determine
the value(s) of x where it converges or diverges.
Note that P(0) = a0.
Power Series
P(x) =
7. A power series P(x) is a "polynomial" in x of infinitely
many terms. That is,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….
Example: A. Let P(x) = 1 + x + x2 + x3 + x4 …
From the theorem of geometric series, given x = r,
I. P(r) = 1 + r + r2 + r3 + r4 + … converges if | r | < 1.
II. P(r) diverges if | r | > 1.
The basic question for a power series is to determine
the value(s) of x where it converges or diverges.
Note that P(0) = a0.
Power Series
P(x) =
8. A power series P(x) is a "polynomial" in x of infinitely
many terms. That is,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….
Example: A. Let P(x) = 1 + x + x2 + x3 + x4 …
From the theorem of geometric series, given x = r,
I. P(r) = 1 + r + r2 + r3 + r4 + … converges if | r | < 1.
II. P(r) diverges if | r | ≥ 1.
The basic question for a power series is to determine
the value(s) of x where it converges or diverges.
Note that P(0) = a0.
Power Series
P(x) =
0
–1 converges (absolutely) 1 divergesdiverges
9. Example: B. Let P(x) = 1 +
Power Series
x x2
+
2
+ x3
3
+ x4
4
+..
10. Example: B. Let P(x) = 1 +
Use the ceiling theorem:
Power Series
x x2
+
2
+ x3
3
+ x4
4
+..
P(x) = 1 +|x|
|x|2
+
2
+
3
+
4
+..
|x|3 |x|4
1 +|x| + |x2| + |x3| + |x4| +.. …
vI vI vI vI vI
11. Example: B. Let P(x) = 1 +
Use the ceiling theorem:
Power Series
x x2
+
2
+ x3
3
+ x4
4
+..
P(x) = 1 +|x|
|x|2
+
2
+
3
+
4
+..
|x|3 |x|4
1 +|x| + |x2| + |x3| + |x4| +.. …
vI vI vI vI vI
and 1 +|x| + |x2| + |x3| + |x4| +… converges for | x | < 1,
we conclude that
12. Example: B. Let P(x) = 1 +
Use the ceiling theorem:
Power Series
x x2
+
2
+ x3
3
+ x4
4
+..
P(x) = 1 +|x|
|x|2
+
2
+
3
+
4
+..
|x|3 |x|4
1 +|x| + |x2| + |x3| + |x4| +.. …
vI vI vI vI vI
and 1 +|x| + |x2| + |x3| + |x4| +… converges for | x | < 1,
we conclude that
P(x) = 1 +|x|
|x|2
+
2
+
3
+
4
+..
|x|3 |x|4
also converges absolutely for |x| < 1.
14. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
For x = ±1:
15. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
P(–1) = 1 – is basically the
alternating harmonic series. It converges conditionally.
1 1 –
2
+ 1
3
+ 1
4 – ..
For x = ±1:
16. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
For x = | r | > 1,
P(–1) = 1 – is basically the
alternating harmonic series. It converges conditionally.
1 1 –
2
+ 1
3
+ 1
4 – ..
For x = ±1:
17. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
For x = | r | > 1, the general terms of P(r)
P(–1) = 1 – is basically the
alternating harmonic series. It converges conditionally.
1 1 –
2
+ 1
3
+ 1
4 – ..
For x = ±1:
rn
n ∞ as n ∞
18. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
For x = | r | > 1, the general terms of P(r)
P(–1) = 1 – is basically the
alternating harmonic series. It converges conditionally.
1 1 –
2
+ 1
3
+ 1
4 – ..
For x = ±1:
rn
n ∞ as n ∞ (verify this by L'Hopital's rule).
19. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
For x = | r | > 1, the general terms of P(r)
P(r) = 1 + r r2
+
2
+
3
+
4
+.. diverges.r3 r4
P(–1) = 1 – is basically the
alternating harmonic series. It converges conditionally.
1 1 –
2
+ 1
3
+ 1
4 – ..
For x = ±1:
rn
n ∞ as n ∞ (verify this by L'Hopital's rule).
Hence
20. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
For x = | r | > 1, the general terms of P(r)
P(r) = 1 + r r2
+
2
+
3
+
4
+.. diverges.r3 r4
P(–1) = 1 – is basically the
alternating harmonic series. It converges conditionally.
1 1 –
2
+ 1
3
+ 1
4 – ..
For x = ±1:
rn
n ∞ as n ∞ (verify this by L'Hopital's rule).
Hence
We summarize the result using the following picture:
21. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
For x = | r | > 1, the general terms of P(r)
P(r) = 1 + r r2
+
2
+
3
+
4
+.. diverges.r3 r4
P(–1) = 1 – is basically the
alternating harmonic series. It converges conditionally.
1 1 –
2
+ 1
3
+ 1
4 – ..
For x = ±1:
rn
n ∞ as n ∞ (verify this by L'Hopital's rule).
Hence
We summarize the result using the following picture:
0
–1 converges (absolutely) 1 divergesdiverges
P(x) = Σ
xn
n
22. P(1) = 1 + is basically the
harmonic series, so it diverges.
Power Series
1 1+
2
+ 1
3
+ 1
4
+..
For x = | r | > 1, the general terms of P(r)
P(r) = 1 + r r2
+
2
+
3
+
4
+.. diverges.r3 r4
P(–1) = 1 – is basically the
alternating harmonic series. It converges conditionally.
1 1 –
2
+ 1
3
+ 1
4 – ..
For x = ±1:
rn
n ∞ as n ∞ (verify this by L'Hopital's rule).
Hence
We summarize the result using the following picture:
0
–1 converges (absolutely) 1
converges conditionally diverges
divergesdiverges
P(x) = Σ
xn
n
23. Theorem: Given a power series , one of the
following must be true:
Power Series
Σn=0
P(x)
∞
24. Theorem: Given a power series , one of the
following must be true:
I. P(x) converges at x = 0, diverges everywhere else.
Power Series
ΣP(x)
∞
n=0
25. Theorem: Given a power series , one of the
following must be true:
I. P(x) converges at x = 0, diverges everywhere else.
Power Series
ΣP(x)
∞
0
converges at 0 divergesdiverges
n=0
x
26. Theorem: Given a power series , one of the
following must be true:
I. P(x) converges at x = 0, diverges everywhere else.
Power Series
ΣP(x)
∞
II. P(x) converges (absolutely) for | x | < R for some
positive R,
0
converges at 0 divergesdiverges
n=0
x
27. Theorem: Given a power series , one of the
following must be true:
I. P(x) converges at x = 0, diverges everywhere else.
Power Series
ΣP(x)
∞
II. P(x) converges (absolutely) for | x | < R for some
positive R, diverges for | x | > R,
0
converges at 0 divergesdiverges
n=0
converges (absolutely) R divergesdiverges
x
x
0
–R
28. Theorem: Given a power series , one of the
following must be true:
I. P(x) converges at x = 0, diverges everywhere else.
Power Series
ΣP(x)
∞
II. P(x) converges (absolutely) for | x | < R for some
positive R, diverges for | x | > R,
and at x = ±R, the series may converge or diverge.
0
converges at 0 divergesdiverges
n=0
converges (absolutely) R divergesdiverges
x
x
0? ?
–R
29. Theorem: Given a power series , one of the
following must be true:
I. P(x) converges at x = 0, diverges everywhere else.
Power Series
ΣP(x)
∞
II. P(x) converges (absolutely) for | x | < R for some
positive R, diverges for | x | > R,
and at x = ±R, the series may converge or diverge.
R is called the radius of convergence.
0
converges at 0 divergesdiverges
0
–R converges (absolutely) R
? ?
divergesdiverges
n=0
x
x
30. Theorem: Given a power series , one of the
following must be true:
I. P(x) converges at x = 0, diverges everywhere else.
Power Series
ΣP(x)
∞
II. P(x) converges (absolutely) for | x | < R for some
positive R, diverges for | x | > R,
and at x = ±R, the series may converge or diverge.
R is called the radius of convergence.
0
converges at 0 divergesdiverges
0
converges (absolutely) R
? ?
divergesdiverges
III. P(x) converges everywhere.
0
convergesconverges
n=0
x
x
x–R
31. Power Series
We may calculate the radius of convergence using
the ratio test or the root test.
32. Power Series
We may calculate the radius of convergence using
the ratio test or the root test.
Example C.
Discuss the convergence of P(x) = Σn=0
∞
xn
n! .
33. Power Series
Since the coefficients are factorials, use the ratio test:
We may calculate the radius of convergence using
the ratio test or the root test.
Example C.
Discuss the convergence of P(x) = Σn=0
∞
xn
n! .
34. Power Series
Since the coefficients are factorials, use the ratio test:
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 ... convergesP(x) =
for x if |an+1xn+1|*
|anxn|
1
< 1 as n ∞.
We may calculate the radius of convergence using
the ratio test or the root test.
Example C.
Discuss the convergence of P(x) = Σn=0
∞
xn
n! .
35. Power Series
Since the coefficients are factorials, use the ratio test:
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 ... convergesP(x) =
for x if |an+1xn+1|*
|anxn|
1
< 1 as n ∞.
|an+1xn+1|*
|anxn|
1
=
xn+1
(n+1)!
*
n!
xn
We may calculate the radius of convergence using
the ratio test or the root test.
Example C.
Discuss the convergence of P(x) = Σn=0
∞
xn
n! .
36. Power Series
Since the coefficients are factorials, use the ratio test:
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 ... convergesP(x) =
for x if |an+1xn+1|*
|anxn|
1
< 1 as n ∞.
|an+1xn+1|*
|anxn|
1
=
xn+1
(n+1)!
*
n!
xn =
x
n+1
We may calculate the radius of convergence using
the ratio test or the root test.
Example C.
Discuss the convergence of P(x) = Σn=0
∞
xn
n! .
37. Power Series
Since the coefficients are factorials, use the ratio test:
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 ... convergesP(x) =
for x if |an+1xn+1|*
|anxn|
1
< 1 as n ∞.
|an+1xn+1|*
|anxn|
1
=
xn+1
(n+1)!
*
n!
xn =
x
n+1
We may calculate the radius of convergence using
the ratio test or the root test.
As n ∞, for any real number x,
x
n+1 0 < 1.
Example C.
Discuss the convergence of P(x) = Σn=0
∞
xn
n! .
38. Power Series
Since the coefficients are factorials, use the ratio test:
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 ... convergesP(x) =
for x if |an+1xn+1|*
|anxn|
1
< 1 as n ∞.
|an+1xn+1|*
|anxn|
1
=
xn+1
(n+1)!
*
n!
xn =
x
n+1
We may calculate the radius of convergence using
the ratio test or the root test.
As n ∞, for any real number x,
x
n+1 0 < 1.
Example C.
Discuss the convergence of P(x) = Σn=0
∞
xn
n! .
39. Power Series
Since the coefficients are factorials, use the ratio test:
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 ... convergesP(x) =
for x if |an+1xn+1|*
|anxn|
1
< 1 as n ∞.
|an+1xn+1|*
|anxn|
1
=
xn+1
(n+1)!
*
n!
xn =
x
n+1
Example C.
Discuss the convergence of P(x) = Σn=0
∞
xn
n! .
We may calculate the radius of convergence using
the ratio test or the root test.
As n ∞, for any real number x,
x
n+1 0 < 1.
Hence the series converegs for all numbers x.
41. Power Series
By the root test, for any real number x,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….P(x) =
converges if lim |anxn| < 1, diverges if it's > 1.n
n∞
Example D.
Discuss the convergence of P(x) = Σn=0
∞
xn
2n
42. Power Series
By the root test, for any real number x,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….P(x) =
converges if lim |anxn| < 1, diverges if it's > 1.
anxn = xn
2n,
n
n∞
Example D.
Discuss the convergence of P(x) = Σn=0
∞
xn
2n
43. Power Series
By the root test, for any real number x,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….P(x) =
converges if lim |anxn| < 1, diverges if it's > 1.
anxn = xn
2n,
so the series converges if
n
n
n∞
lim |xn/2n|n∞
Example D.
Discuss the convergence of P(x) = Σn=0
∞
xn
2n
44. Power Series
By the root test, for any real number x,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….P(x) =
converges if lim |anxn| < 1, diverges if it's > 1.
anxn = xn
2n,
so the series converges if
n
n
n∞
lim |xn/2n| = |x/2| < 1n∞
Example D.
Discuss the convergence of P(x) = Σn=0
∞
xn
2n
45. Power Series
By the root test, for any real number x,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….P(x) =
converges if lim |anxn| < 1, diverges if it's > 1.
anxn = xn
2n,
so the series converges if
n
n
n∞
lim |xn/2n| = |x/2| < 1 or | x | < 2.n∞
Example D.
Discuss the convergence of P(x) = Σn=0
∞
xn
2n
46. Power Series
By the root test, for any real number x,
Σn=0
∞
anxn = a0 + a1x + a2x2 + a3x3 + a4x4 ….P(x) =
converges if lim |anxn| < 1, diverges if it's > 1.
anxn = xn
2n,
and diverges if | x | > 2,
and the radius of convergence R = 2.
so the series converges if
n
n
n∞
lim |xn/2n| = |x/2| < 1 or | x | < 2.n∞
Example D.
Discuss the convergence of P(x) = Σn=0
∞
xn
2n
48. Power Series
Finally, we clarify what happen at the boundary points
x = ±2.
P(2) = Σn=0
2n
2n = 1 + 1 + 1 + … so it diverges at x = 2.
49. Power Series
Finally, we clarify what happen at the boundary points
x = ±2.
P(2) = Σn=0
2n
2n = 1 + 1 + 1 + … so it diverges at x = 2.
P(–2) =Σn=0
(–2)n
2n
= 1 – 1 + 1 – 1.. so it diverges at x = –2.
50. Power Series
Finally, we clarify what happen at the boundary points
x = ±2.
P(2) = Σn=0
2n
2n = 1 + 1 + 1 + … so it diverges at x = 2.
P(–2) =Σn=0
(–2)n
2n
= 1 – 1 + 1 – 1.. so it diverges at x = –2.
So P(x) converges for | x | < 2, diverges for | x | ≥ 2.
51. Power Series
Finally, we clarify what happen at the boundary points
x = ±2.
P(2) = Σn=0
2n
2n = 1 + 1 + 1 + … so it diverges at x = 2.
P(–2) =Σn=0
(–2)n
2n
= 1 – 1 + 1 – 1.. so it diverges at x = –2.
So P(x) converges for | x | < 2, diverges for | x | ≥ 2.
We may shift a power series by replacing x by (x – a).
52. Power Series
Finally, we clarify what happen at the boundary points
x = ±2.
P(2) = Σn=0
2n
2n = 1 + 1 + 1 + … so it diverges at x = 2.
P(–2) =Σn=0
(–2)n
2n
= 1 – 1 + 1 – 1.. so it diverges at x = –2.
So P(x) converges for | x | < 2, diverges for | x | ≥ 2.
We may shift a power series by replacing x by (x – a).
Σn=0
∞
an(x – a)n = a0 + a1(x – a) + a2(x – a)2 + a3(x – a)3 +..
∞
is said to be a power series centered at x = a.
A power defined in the form of
53. Power Series
Finally, we clarify what happen at the boundary points
x = ±2.
P(2) = Σn=0
2n
2n = 1 + 1 + 1 + … so it diverges at x = 2.
P(–2) =Σn=0
(–2)n
2n
= 1 – 1 + 1 – 1.. so it diverges at x = –2.
So P(x) converges for | x | < 2, diverges for | x | ≥ 2.
We may shift a power series by replacing x by (x – a).
Σn=0
∞
an(x – a)n = a0 + a1(x – a) + a2(x – a)2 + a3(x – a)3 +..
The power series Σn=0
∞
anxn
is said to be a power series centered at x = a.
is the special case of
Σn=0
∞
an(x – a)n that’s center at a = 0.
A power defined in the form of
54. Theorem: Given a power series , one of the
following must be true:
Power Series
Σn=0
∞
an(x – a)n
55. Theorem: Given a power series , one of the
following must be true:
I. It converges at x = a, diverges everywhere else.
Power Series
a
converges at a divergesdiverges
Σn=0
∞
an(x – a)n
56. Theorem: Given a power series , one of the
following must be true:
I. It converges at x = a, diverges everywhere else.
Power Series
II. For some positive R, it converges (absolutely) for
| x – a | < R, diverges for | x – a | > R,
and at x = a±R, the series may converge or diverge.
(a – R, a + R) is called the interval of convergence.
a
converges at a divergesdiverges
a
a – R converges (absolutely) a + R
? ?
divergesdiverges
Σn=0
∞
an(x – a)n
57. Theorem: Given a power series , one of the
following must be true:
I. It converges at x = a, diverges everywhere else.
Power Series
II. For some positive R, it converges (absolutely) for
| x – a | < R, diverges for | x – a | > R,
and at x = a±R, the series may converge or diverge.
(a – R, a + R) is called the interval of convergence.
a
converges at a divergesdiverges
a
a – R converges (absolutely) a + R
? ?
divergesdiverges
III. It converges everywhere.
a
convergesconverges
Σn=0
∞
an(x – a)n
59. Power Series
Use the ratio test,
Example E.
Discuss the convergence of P(x) = Σn=1
∞
Ln(n)(x – 3)n
n
60. Power Series
Use the ratio test,
|an+1 (x –a)n+1|*
|an(x – a)n|
1
Example E.
Discuss the convergence of P(x) = Σn=1
∞
Ln(n)(x – 3)n
n
61. Example E.
Discuss the convergence of P(x) =
Power Series
Σn=1
∞
Ln(n)(x – 3)n
n
Use the ratio test,
|an+1 (x –a)n+1|*
|an(x – a)n|
1
= Ln(n+1)(x – 3)n+1
n+1 *
Ln(n)(x – 3)n
n
62. Power Series
Use the ratio test,
|an+1 (x –a)n+1|*
|an(x – a)n|
1
= Ln(n+1)(x – 3)n+1
n+1 *
Ln(n)(x – 3)n
n
= n * Ln(n+1) (x – 3)
(n+1) * Ln(n)
Example E.
Discuss the convergence of P(x) = Σn=1
∞
Ln(n)(x – 3)n
n
63. Power Series
Use the ratio test,
|an+1 (x –a)n+1|*
|an(x – a)n|
1
= Ln(n+1)(x – 3)n+1
n+1 *
Ln(n)(x – 3)n
n
= n * Ln(n+1) (x – 3)
(n+1) * Ln(n)
, as n ∞
Example E.
Discuss the convergence of P(x) = Σn=1
∞
Ln(n)(x – 3)n
n
64. Power Series
Use the ratio test,
|an+1 (x –a)n+1|*
|an(x – a)n|
1
= Ln(n+1)(x – 3)n+1
n+1 *
Ln(n)(x – 3)n
n
= n * Ln(n+1) (x – 3)
(n+1) * Ln(n)
, as n ∞
1 1
Example E.
Discuss the convergence of P(x) = Σn=1
∞
Ln(n)(x – 3)n
n
65. Power Series
Use the ratio test,
|an+1 (x –a)n+1|*
|an(x – a)n|
1
= Ln(n+1)(x – 3)n+1
n+1 *
Ln(n)(x – 3)n
n
= n * Ln(n+1) (x – 3)
(n+1) * Ln(n)
, as n ∞ we get |x – 3|
1 1
Example E.
Discuss the convergence of P(x) = Σn=1
∞
Ln(n)(x – 3)n
n
66. Power Series
Use the ratio test,
|an+1 (x –a)n+1|*
|an(x – a)n|
1
= Ln(n+1)(x – 3)n+1
n+1 *
Ln(n)(x – 3)n
n
= n * Ln(n+1) (x – 3)
(n+1) * Ln(n)
, as n ∞ we get |x – 3|
1 1
The series converges if |x – 3| < 1
Example E.
Discuss the convergence of P(x) = Σn=1
∞
Ln(n)(x – 3)n
n
67. Power Series
Use the ratio test,
|an+1 (x –a)n+1|*
|an(x – a)n|
1
= Ln(n+1)(x – 3)n+1
n+1 *
Ln(n)(x – 3)n
n
= n * Ln(n+1) (x – 3)
(n+1) * Ln(n)
, as n ∞ we get |x – 3|
1 1
The series converges if |x – 3| < 1 so the interval
of convergence is (2, 4).
Example E.
Discuss the convergence of P(x) = Σn=1
∞
Ln(n)(x – 3)n
n
71. P(4) =
Power Series
Σn=1
∞ Ln(n)(4 – 3)n
n = Σn=1
∞
Ln(n)
n > Σn=2
∞
1
n = ∞
For the end points of the interval:
72. P(4) =
Power Series
Σn=1
∞ Ln(n)(4 – 3)n
n
Σn=1
∞ Ln(n)(2 – 3)n
n
= Σn=1
∞
Ln(n)
n > Σn=2
∞
1
n = ∞
P(2) =
For the end points of the interval:
73. P(4) =
Power Series
Σn=1
∞ Ln(n)(4 – 3)n
n
Σn=1
∞ Ln(n)(2 – 3)n
n
= Σn=1
∞
Ln(n)
n > Σn=2
∞
1
n = ∞
P(2) =
For the end points of the interval:
= Σn=1
∞
(–1)nLn(n)
n
74. P(4) =
Power Series
Σn=1
∞ Ln(n)(4 – 3)n
n
Σn=1
∞ Ln(n)(2 – 3)n
n
= Σn=1
∞
Ln(n)
n > Σn=2
∞
1
n = ∞
P(2) =
For the end points of the interval:
= Σn=1
∞
(–1)nLn(n)
n
is a decreasing alternating series with term 0,
so it converges.
75. P(4) =
Power Series
Σn=1
∞ Ln(n)(4 – 3)n
n
Σn=1
∞ Ln(n)(2 – 3)n
n
= Σn=1
∞
Ln(n)
n
is a decreasing alternating series with term 0,
so it converges. However it converges conditionally.
> Σn=2
∞
1
n = ∞
P(2) =
For the end points of the interval:
= Σn=1
∞
(–1)nLn(n)
n
76. P(4) =
Power Series
Σn=1
∞ Ln(n)(4 – 3)n
n
Σn=1
∞ Ln(n)(2 – 3)n
n
= Σn=1
∞
Ln(n)
n > Σn=2
∞
1
n = ∞
P(2) =
For the end points of the interval:
= Σn=1
∞
(–1)nLn(n)
n
Hence we have the following graph for convergence:
3
2 converges (absolutely) 4
converges conditionally
divergesdiverges
diverges
is a decreasing alternating series with term 0,
so it converges. However it converges conditionally.