http://www.ck12.org Chapter 8. Infinite Series
8.6 Power Series
Power Series and Convergence
Definition
(Power Series)
APower Seriesis a series of the form(PS1)∑∞n= 0 anxn=a 0 +a 1 x+a 2 x^2 +a 3 x^3 +...
wherexis a variable and thean’s are constants (in our case, real numbers) called thecoefficientsof the series.
The summation sign∑is a compact and convenient shorthand notation. Readers unfamiliar with the notation might
want to write out the detail a few times to get used to it.
Power series are a generalization of polynomials, potentially with infinitely many terms. As observed, the indicesn
ofanare non-negative, so no negative integral exponents ofx, e.g.^1 xappear in a power series.
More generally, a series of the form
(PS2)∑∞n= 0 an(x−x 0 )n=a 0 +a 1 (x−x 0 )+a 2 (x−x 0 )^2 +a 3 (x−x 0 )^3 +...
is called a power series in(x−x 0 )( 1 = (x−x 0 )^0 )or a power seriescenteredatx 0 ((PS1) represents a series centered
atx=0).
Given any value ofx, a power series ((PS1) and (PS2)) is a series of numbers. The first question is:
Is the power series (as in (PS1) or (PS2)) a function ofx?
Since the series is always defined atx=0 (resp.x=x 0 ), the question becomes:
For what value ofxis a power series convergent?
The answers are known for some series. Convergence tests could be applied on some others.
Example 1Letr 6 =0 andx 0 be real.
∑∞n= 0 (r(x−x 0 ))n= 1 +r(x−x 0 )+r^2 (x−x 0 )^2 +r^3 (x−x 0 )^3 +...is absolutely convergent and equals 1 −r(^1 x−x 0 )for
|r(x−x 0 )|<1, i.e.|x−x 0 |<∣∣^1 r∣∣, and diverges otherwise.Letr=1;x 0 = 0 .Then∑xnis the power series for 1 −^1 xon(− 1 , 1 ). Letr=−1;x 0 = 2 .Then∑(− 1 )n(x− 2 )nis
the power series for 1 −(− 11 )(x− 2 )=x−^11 on( 1 , 3 ). So∑xnand∑(− 1 )n−^1 (x− 2 )nare the power series for the same
function 1 −^1 xbut on different intervals. There is a more detailed discussion in §8.7.
Example 2∑∞n= 0 x^2 nis absolutely convergent for|x|<1 by Comparison Test (against∑∞n= 0 xn) and diverges for|x|≥ 1
by the Test for Divergence.
Exercises
- Write a power series∑∞n= 0 an(x+ 2 )ncentered atx=−2 for the same function 1 −^1 xin Example 6.1.1. On what
interval does equality hold?
Hint: Substitutey=x+2 in 1 −^1 x.