How do you test a power series for convergence?

1 Answer
Wataru
Sep 26, 2014

Since the convergence of a power series depend on the value of #x#, so the question should be "For which value of #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 #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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