Question

How do you test a power series for convergence?

Answer 1

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)`.