# What is interval of convergence for a Power Series?

Sep 10, 2014

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 ${\sum}_{n = 0}^{\infty} \frac{{x}^{n}}{n}$.
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}
$= | x | \cdot 1 = | x | < 1 R i g h t a r r o w - 1 < x < 1$,
which means that the power series converges at least on $\left(- 1 , 1\right)$.

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
${\sum}_{n = 0}^{\infty} {\left(- 1\right)}^{n} / n$,
which is convergent. So, $x = 1$ should be included.

If $x = 1$, the power series becomes the harmonic series
${\sum}_{n = 0}^{\infty} \frac{1}{n}$,
which is divergent. So, $x = 1$ should be excluded.

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