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
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
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.
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.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
4. ▶ Zenos Paradox. If the distance between the person and the
wall is 1 then intuitively we know
1
2
+
1
4
+
1
8
+
1
16
+ . . . +
1
2n
+ . . . = 1
▶ We can write out π as
π = 3.14159 26535 89793 . . .
π = 3 +
1
10
+
4
102
+
1
103
+
5
104
+ . . .
▶ Though we can’t literally add an infinite number of terms, the
more we add, the closer we get to the actual value of π.
5. Infinite Series
▶ An infinite series (or just a series) is the sum
a1 + a2 + . . . + an + . . . =
∞
X
n=1
an =
X
an
of an infinite sequence {an}∞
n=1.
6. Sum of an Infinite Series - Zeno’s Paradox
▶ Consider the example given in Zeno’s paradox.
▶ We know that
∞
X
n=1
1
2n
= 1
intuitively. But how do we prove it mathematically?
▶ Let sn =
n
X
i=1
1
2i
=
1
2
+
1
4
+
1
8
+ . . . +
1
2n−1
+
1
2n
▶ this is called the nth
partial sum of the series
∞
X
n=1
1
2n
7. Sum of an Infinite Series - Zeno’s Paradox
2sn = 2
n
X
i=1
1
2i
=
2
2
+
2
22
+
2
23
+ . . . +
2
2n−1
+
2
2n
= 1 +
1
2
+
1
4
+ . . . +
1
2n−1
= 1 +
n−1
X
i=1
1
2i
= 1 +
n
X
i=1
1
2i
−
1
2n
=⇒
2sn = 1 + sn −
1
2n
sn = 1 −
1
2n
Now,
lim
n→∞
sn = lim
n→∞
1 −
1
2n
= 1 ✓
8. Sum of an Infinite Series
We take this general approach as follows:
▶ Given a series
∞
X
n=1
an = a1 + a2 + . . . + an + . . .
let sn denote its nth partial sum:
sn =
n
X
i=1
ai = a1 + a2 + . . . + an−1 + an
9. Sum of an Infinite Series
▶ If the sequence of terms an is convergent and the sequence of
partial sums is convergent, ie,
lim
n→∞
n
X
i=1
an = lim
n→∞
sn = s
exists as a real number, then the series
∞
X
n=1
an is convergent
and we write
∞
X
n=1
an = s
▶ The number s is called the sum of the series.
▶ If the sequence sn is divergent then the series is called
divergent.
10. Sum of an Infinite Series
▶ Remark: Any series can be written
∞
X
n=1
an =
N
X
n=1
an +
∞
X
n=N+1
aN
▶ Taking the limit of both sides,
lim
N→∞
∞
X
n=1
an = lim
N→∞
N
X
n=1
an + lim
N→∞
∞
X
n=N+1
aN
∞
X
n=1
an = lim
N→∞
N
X
n=1
an
= lim
N→∞
sN
lim
n→∞
n
X
i=1
ai =
∞
X
n=1
an
11. Example 1
Suppose we know that the nth partial sum of the series
∞
X
n=1
an is
sn =
2n
3n + 5
To determine if the series
∞
X
n=1
an converges, we take the limit of sn.
lim
n→∞
sn = lim
n→∞
2n
3n + 5
= lim
n→∞
2n
n
3n
n + 5
n
= lim
n→∞
2
3 + 5
n
=
2
3
So the series
P∞
n=1 an converges to 2
3 , ie,
P∞
n=1 an = 2
3 .
12. Example 2
1/2
Suppose we want to determine whether the following series
converges, and find the sum if possible.
∞
X
n=1
1
n(n + 1)
▶ This time we must find a formula for the partial sum:
sn =
n
X
i=1
1
i(i + 1)
▶ We can write 1
i(i+1) = i+1−i
i(i+1) = i+1
i(i+1) − i
i(i+1) = 1
i − 1
i+1 so
sn =
n
X
i=1
1
i
−
1
i + 1
13. Example 2
2/2
sn =
n
X
i=1
1
i
−
1
i + 1
=
1
1
−
1
2
+
1
2
−
1
3
+
1
3
−
1
4
+ . . . +
1
n
−
1
n + 1
= 1 −
1
n + 1
▶ A sum which has terms that cancel in pairs is called
telescoping.
▶ The sum collapses into two terms like a pirates collapsing
telescope.
lim
n→∞
sn = lim
n→∞
1 −
1
n + 1
= 1 =⇒
∞
X
n=1
1
n(n + 1)
= 1
15. Geometric Series
▶ A geometric series is a series of the form
∞
X
n=1
arn−1
= a + ar + ar2
+ . . . + arn−1
+ . . . (a ̸= 0)
eg, The series in Zenos paradox is geometric with a = r = 1
2
eg,
∞
X
n=1
7n
5n+1
is a geometric series with a = 1
5 and r = 7
5 because
we can write
7n
5n+1
=
7n
5 · 5n
=
1
5
·
7n
5n
=
1
5
7
5
n
16. ▶ Now let’s determine for which values of r the geometric series
∞
X
n=1
arn−1
converges.
▶ If r = 1 then
sn =
n
X
i=1
a = a + a + a + . . . + a = na
=⇒ lim
n→∞
sn = lim
n→∞
na
= a lim
n→∞
n =
(
∞ if a 0
−∞ if a 0
▶ Note: infinite series of a nonzero constant is always divergent
∞
X
n=1
a = a + a + a + . . . + a + . . . → ±∞
17. ▶ If r ̸= 1 we have
sn = a + ar + ar2
+ . . . + arn−2
+ arn−1
rsn = ar + ar2
+ ar3
+ . . . + arn−1
| {z }
sn−a
+arn
= sn − a + arn
=⇒
rsn − sn = arn
− a
sn(r − 1) = a(rn
− 1) =⇒
sn =
a(1 − rn)
1 − r
18. ▶ Now taking the limit of sn we have
lim
n→∞
sn = lim
n→∞
a(1 − rn)
1 − r
=
a
1 − r
lim
n→∞
(1 − rn
)
=
a
1 − r
−
a
1 − r
lim
n→∞
rn
▶ Recall from 11.1 that {rn} is convergent if −1 r ≤ 1 and
divergent for all other values of r.
▶ In particular
lim
n→∞
rn
=
0 if − 1 r 1
1 if r = 1
Diverges for all other values of r
19. Geometric Series Convergence
▶ So for −1 r 1,
lim
n→∞
sn =
a
1 − r
−
a
1 − r
lim
n→∞
rn
=
a
1 − r
∞
X
n=1
arn−1
=
a
1 − r
if |r| 1 and divergent for all other values of r
20. Example 3
5 −
10
3
+
20
9
−
40
27
+ . . .
▶ To determine if the series converges and find its sum if
possible, we notice it is geometric and write it in the form
P∞
n=1 arn−1.
5 −
10
3
+
20
9
−
40
27
+ . . . = 5
1 −
2
3
+
4
9
−
8
27
− . . .
= 5
∞
X
n=1
−
2
3
n−1
=
∞
X
n=1
5
−
2
3
n−1
▶ a = 5, r = −2
3 and since |r| = 2
3 1, the series converges to
s =
5
1 + 2
3
=
5
5
3
= 3
22. Example 5
1/2
▶ A rational number q is a number that can be written in the
form q = m
n where m and n are integers and n ̸= 0.
▶ Any number with a repeated decimal representation is a
rational number because it can be written in this form.
▶ As an example, let’s write the number
2.34 = 2.3434343434 . . . as a ratio of integers.
2.34 = 2 +
3
10
+
4
100
+ +
3
1, 000
+
4
10, 000
+
3
105
+
4
106
+ . . .
= 2 + 3
1
10
+
1
1, 000
+
1
105
+ . . .
+ 4
1
100
+
1
10, 000
+
1
106
+ . . .
= 2 + 3
1
101
+
1
103
+
1
105
+ . . .
+ 4
1
102
+
1
104
+
1
106
+ . . .
= 2 + 3
∞
X
n=1
1
102n−1
+ 4
∞
X
n=1
1
102n
24. Example 6
1/2
Find the sum of the series
∞
X
n=0
xn
where |x| 1
∞
X
n=0
xn
= x0
+ x1
+ x2
+ x3
+ . . .
= 1 + x + x2
+ x3
+ . . .
= 1 +
∞
X
n=1
xn
= 1 +
∞
X
n=1
x · xn−1
25. Example 6
2/2
1 +
∞
X
n=1
x · xn−1
| {z }
a=x, r=x
= 1 +
x
1 − x
| {z }
a
1−r
=
1 − x
1 − x
+
x
1 − x
=
1
1 − x
▶ This example demonstrates that on the interval (−1, 1), the
function f (x) =
1
1 − x
has the power series representation
f (x) =
∞
X
n=0
xn
.
27. ▶ A series
∞
X
n=1
an is divergent if its sequence of partial sums
sn =
n
X
i=1
ai is divergent.
28. Harmonic Series
Show that the harmonic series
∞
X
n=1
1
n
is divergent.
▶ The harmonic series is an important series whose name derives
from the concept of overtones in music.
▶ It will be difficult to find a simple formula for the nth partial
sum.
▶ Instead we will show that the 2nth partial sum s2n , is divergent.
▶ Writing out the first few terms of s2n we will observe that
s2n 1 + n
2 for each n.
31. Harmonic Series
s24 = s16 = s8 +
1
9
+
1
10
+
1
11
+ . . . +
1
16
(n=4)
s8 +
1
16
+
1
16
+
1
16
+ . . . +
1
16
= s8 +
8
16
1 +
4
2
since s8 1 +
3
2
▶ In general s2n 1 + n
2 for each n.
lim
n→∞
s2n 1 + lim
n→∞
n
2
= ∞
▶ Since s2n diverges, that means sn must diverge as well.
▶ Thus the harmonic series diverges.
32. Divergent Series
▶ Finding a formula for the nth partial sum of a series can be
quite challenging.
▶ In many cases, taking the limit of the partial sum won’t be the
most efficient method to determine whether a series converges
or diverges.
▶ In order for the series
∞
X
n=1
an to converge, it’s sequence of
terms an must converge as well.
▶ In fact, the next theorem tells us something even stronger.
34. Test for Divergence
Theorem 6: If the series
∞
X
n=1
an is convergent, then lim
n→∞
an = 0.
▶ WARNING: The converse of this theorem is false.
▶ ie, lim
n→∞
an = 0 does not imply that the series
∞
X
n=1
an converges.
▶ The harmonic series is an example of when the converse fails
since limn→∞
1
n = 0 but
P 1
n diverges.
▶ However, the contrapositive statement of the theorem, which
is equivalent to the theorem, is called the Test for Divergence
and will be very useful.
Test for Divergence : lim
n→∞
an ̸= 0 =⇒
∞
X
n=1
an is divergent.
38. Theorem: If
P
an =
P
bn are convergent series, then so are the
series
P
can,
P
an +
P
bn, and
P
an −
P
bn
(i)
P
can = c
P
an
(ii)
P
(an + bn) =
P
an +
P
bn
(iii)
P
(an − bn) =
P
an −
P
bn
40. Example 8
2/2
▶ From example 1 we have,
∞
X
n=1
1
n(n + 1)
= 1
▶ And from the geometric series theorem,
∞
X
n=1
1
2n
= 1
∞
X
n=1
3
n(n + 1)
+
1
2n
= 3
∞
X
n=1
1
n(n + 1)
+
∞
X
n=1
1
2n
= 3 + 1
= 4