How do you test a power series for convergence?

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How do you test a power series for convergence?

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Answer 1 · 2014-09-26T14:20:31

Since the convergence of a power series depend on the value of $x$ $x$ , so the question should be "For which value of $x$ $x$ does a power series converges?" For most cases, the ratio test will do the trick.

Here is an example.

The interval of convergence of a power series is the set of all x-values for which the power series converges.

Let us find the interval of convergence of $\infty \sum n = 0 \frac{x^{n}}{n}$ $sum_{n=0}^infty{x^n}/n$ .
By Ratio Test,
$lim_{n to infty}|{a_{n+1}}/{a_n}| =lim_{n to infty}|x^{n+1}/{n+1}cdotn/x^n| =|x|lim_{n to infty}n/{n+1}$
$=|x|cdot 1=|x|<1 Rightarrow -1 < x < 1$ ,
which means that the power series converges at least on $(-1,1)$ .

Now, we need to check its convergence at the endpoints: $x=-1$ and $x=1$ .

If $x=-1$ , the power series becomes the alternating harmonic series
$sum_{n=0}^infty(-1)^n/n$ ,
which is convergent. So, $x=1$ should be included.

If $x=1$ , the power series becomes the harmonic series
$sum_{n=0}^infty1/n$ ,
which is divergent. So, $x=1$ should be excluded.

Hence, the interval of convergence is $[-1,1)$ .

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How do you test a power series for convergence?

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