Step By Step Calculus » 17.5 - Power Series

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Synopsis
A power series about aa is a series of the form
\displaystyle \sum_{i=0}^{\infty} c_i(x-a)^i=c_0+c_1(x-a)+c_2(x-a)^2+\cdots+c_n(x-a)^n+\cdots
\displaystyle \sum_{i=0}^{\infty} c_i(x-a)^i=c_0+c_1(x-a)+c_2(x-a)^2+\cdots+c_n(x-a)^n+\cdots
for each xx where convergence takes place. The interval, that contains all values of xx for which the series is convergent, is called the interval of convergence.
A power series \displaystyle \sum_{i=0}^{\infty} c_i(x-a)^i\displaystyle \sum_{i=0}^{\infty} c_i(x-a)^i satisfies one of the following three:
(i) the series converges only for x=ax=a,
(ii) the series converges for all xx, or
(iii) there exists a positive number RR such that the series converges when |x-a|<R|x-a|<R and diverges when |x-a|>R|x-a|>R.
The number RR is called the radius of convergence of the power series. If the series converges for all xx, then R=\inftyR=\infty; if the series only converges for x=ax=a then R=0R=0. One can use the following general procedure to find the interval of convergence. Continue up to Step 2 if only the radius of convergence is required.
Step 1: (Find \mathbf{c_n, a}\mathbf{c_n, a}) Identify the coefficient c_nc_n of n^{\textrm{th}}n^{\textrm{th}}-term and the center aa of the series.
Step 2: (Find \mathbf{R}\mathbf{R}) Calculate RR either using the Ratio Test \displaystyle R=\lim_{n\to\infty}\left|\frac{c_n}{c_{n+1}}\right|\displaystyle R=\lim_{n\to\infty}\left|\frac{c_n}{c_{n+1}}\right| or using the Root Test \displaystyle R=\lim_{n\to\infty}\frac{1}{\sqrt[n]{|c_n|}}\displaystyle R=\lim_{n\to\infty}\frac{1}{\sqrt[n]{|c_n|}}, whichever is easier.
Step 3: (Check Left) Does \displaystyle \sum_{i=1}^{\infty} c_i(-R)^i\displaystyle \sum_{i=1}^{\infty} c_i(-R)^i converge? Use the known methods to check the convergence.
Step 4: (Check Right) Does \displaystyle \sum_{i=1}^{\infty} c_i(R)^i\displaystyle \sum_{i=1}^{\infty} c_i(R)^i converge? Use the known methods to check the convergence.
Step 5: (State Interval of Convergence) This is either (a-R,a+R)(a-R,a+R), or [a-R,a+R)[a-R,a+R), or (a-R, a+R](a-R, a+R], or [a-R,a+R][a-R,a+R].
A convergent power series represents a function, which can be differentiated or integrated by differentiating or integrating the power series term by term. The derived or integrated series will have the same radius of convergence.
Adding or multiplying two convergent power series generates another convergent power series with the same radius of convergence.