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

1. Power SeriesTopis

2. A power series is a series of the form
where x is a variable and the cn’s are constants called the coefficients
of the series.
For each fixed x, the series (1) is a series of constants that we can test
for convergence or divergence.
A power series may converge for some values of x and diverge for other
values of x.

3. The sum of the series is a function
f(x) = c0 + c1x + c2x2 + . . . + cnxn + . . .
whose domain is the set of all x for which the series converges. Notice
that f resembles a polynomial. The only difference is that f has infinitely
many terms.
For instance, if we take cn = 1 for all n, the power series becomes the
geometric series
𝑥=0
∞
xn = 1 + x + x2 + . . . + xn + . . .
which converges when –1 < x < 1 and diverges when |x|  1.

4. More generally, a series of the form
is called a power series in (x – a) or a power series centered at a or a
power series about a.
Notice that in writing out the term corresponding to n = 0 in Equations 1
and 2 we have adopted the convention that
(x – a)0 = 1 even when x = a.
Notice also that when x = a all of the terms are 0 for n  1 and so the
power series (2) always converges when x = a.

5. The following theorem says that this is true in general.

6. I should use the ratio test. It is the test
of choice when testing for convergence
of power series!

7. Power Series
Find the Radius of Convergence and the Interval of Convergence
for the following power series
Example:
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 𝒙 + 𝟑 𝒏
𝟒 𝒏 𝑐 𝑛+1 =
−𝟏 𝒏+𝟏
𝒏 + 𝟏 𝒙 + 𝟑 𝒏+𝟏
𝟒 𝒏+𝟏
Ratio Test
lim
𝒏→∞
−𝟏 𝒏+𝟏
𝒏 + 𝟏 𝒙 + 𝟑 𝒏+𝟏
𝟒 𝒏+𝟏
÷
−𝟏 𝒏
𝒏 𝒙 + 𝟑 𝒏
𝟒 𝒏
lim
𝒏→∞
−𝟏 𝒏 −𝟏 𝒏 + 𝟏 𝒙 + 𝟑 𝒏 𝒙 + 𝟑
𝟒 𝒏 𝟒
∙
𝟒 𝒏
−𝟏 𝒏 𝒏 𝒙 + 𝟑 𝒏
lim
𝒏→∞
−𝟏 𝒏 + 𝟏 𝒙 + 𝟑
𝟒𝒏
=
𝟏
𝟒
𝒙 + 𝟑

8. Section 10.7 – Power Series
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 𝒙 + 𝟑 𝒏
𝟒 𝒏
Ratio Test: Convergence for 𝑳 < 𝟏
lim
𝒏→∞
−𝟏 𝒏 + 𝟏 𝒙 + 𝟑
𝟒𝒏
=
𝟏
𝟒
𝒙 + 𝟑
𝟏
𝟒
𝒙 + 𝟑 < 𝟏
𝒙 + 𝟑 < 𝟒
−𝟒 < 𝒙 + 𝟑 < 𝟒
−𝟕 < 𝒙 < 𝟏
Radius of Convergence
𝑹 = 𝟒
Interval of Convergence:
End points need to be tested.


9. Power Series
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 𝒙 + 𝟑 𝒏
𝟒 𝒏
𝒙 = −𝟕
𝒅𝒊𝒗𝒆𝒓𝒈𝒆𝒏𝒕 𝒂𝒕 𝒙 = −𝟕
= ∞
𝒏 𝒕𝒉
𝒕𝒆𝒓𝒎 𝒕𝒆𝒔𝒕 𝒇𝒐𝒓 𝒅𝒊𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆
−𝟕 𝒄𝒂𝒏𝒏𝒐𝒕 𝒃𝒆 𝒊𝒏𝒄𝒍𝒖𝒅𝒆𝒅 𝒊𝒏
𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 −𝟕 + 𝟑 𝒏
𝟒 𝒏
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 −𝟒 𝒏
𝟒 𝒏
𝒏=𝟎
∞
−𝟏 𝒏
𝒏 −𝟏 𝒏
𝟒 𝒏
𝟒 𝒏
𝒏=𝟎
∞
𝒏
lim
𝒏→∞
𝒏 ≠ 𝟎


10. Power Series
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 𝒙 + 𝟑 𝒏
𝟒 𝒏
𝒙 = 𝟏
𝒅𝒊𝒗𝒆𝒓𝒈𝒆𝒏𝒕 𝒂𝒕 𝒙 = 𝟏
= 𝑫𝑵𝑬
𝒏 𝒕𝒉 𝒕𝒆𝒓𝒎 𝒕𝒆𝒔𝒕 𝒇𝒐𝒓 𝒅𝒊𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆
𝟏 𝒄𝒂𝒏𝒏𝒐𝒕 𝒃𝒆 𝒊𝒏𝒄𝒍𝒖𝒅𝒆𝒅 𝒊𝒏
𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 𝟏 + 𝟑 𝒏
𝟒 𝒏
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 𝟒 𝒏
𝟒 𝒏
𝒏=𝟎
∞
−𝟏 𝒏 𝒏
lim
𝒏→∞
−𝟏 𝒏 𝒏 ≠ 𝟎
𝒏=𝟎
∞
−𝟏 𝒏
𝒏
𝑻𝒉𝒆𝒓𝒆𝒇𝒐𝒓𝒆, 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆 𝒊𝒔:
−𝟕 < 𝒙 < 𝟏


11. 0
1
when | - | the series converges absolutely.x x
k

0
1
when | - | the series diverges.x x
k

0
1
when | - | we don't know.x x
k

0x 0
1
x
k
0
1x
k

Must test
endpoints
separately!

12. 0
0
( )n
n
n
a x x


Theorem: If we have a power series ,
• It may converge only at x=x0.
0x•It may converge for all x.
•It may converge on a finite interval centered at x=x0.
Radius of convergence is 0
Radius of conv. is infinite.
0x 0x R0x R
Radius of conv. is R.

13. 0
0
( )n
n
n
a x x


Theorem: If we have a power series
• It may converge only at x=x0.
0x•It may converge for all x.
•It may converge on a finite interval centered at x=x0.
0x 0x R0x R

14. THANK
YOU