Question

How do you find the interval of convergence for a power series?

Answer 1

Step-1: Concept used:

For the value $\mathrm{x}=\mathrm{c}$, every power series converges absolutely.

There is a parameter $R$, known as the radius of convergence, for every power series that converges at more than one point, such that the series converges absolutely whenever$\mathrm{jx}\mathrm{cj}$ $R$ and diverges when$\mathrm{jx}\mathrm{cj}>\mathrm{R}$. When $\mathrm{jx}\mathrm{cj}=\mathrm{R}$ ,a series may converge or diverge.

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

Step-2: Calculating the interval:

Using ratio test

$\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{{\mathrm{a}}_{\mathrm{n}}+1}{{\mathrm{a}}_{\mathrm{n}}}\right|=\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{{\mathrm{x}}^{\mathrm{n}+1}}{\mathrm{n}+1}.\frac{\mathrm{n}}{{\mathrm{x}}^{\mathrm{n}}}\right|=\left|\mathrm{x}\right|\underset{\mathrm{n}\to \infty }{\lim }\frac{\mathrm{n}}{\mathrm{n}+1}$

$=\left|\mathrm{x}\right|.1=\left|\mathrm{x}\right|<1\Rightarrow -1<\mathrm{x}<1$

This indicates that the power series converges at least on (-1,1).

Checking its convergence at the endpoints $\mathrm{x}=-1$ and $\mathrm{x}=1$.

if $\mathrm{x}=-1$ the power series becomes the alternating harmonic series.

$\sum _{\mathrm{n}=0}^{\infty }\frac{{(-1)}^{\mathrm{n}}}{\mathrm{n}}$

which is convergent. So, $\mathrm{x}=1$ should be included.

If $\mathrm{x}=1$, the power series becomes the harmonic series

$\sum _{\mathrm{n}=0}^{\infty }\frac{1}{\mathrm{n}}$

which is divergent. So $\mathrm{x}=1$ should be excluded.

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

Therefore, the interval of convergence of a power series is $[-1,1)$.