Question

Let $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$be a power series in $x$with a finite radius of convergence $p$. Prove that if the series converges absolutely at either $\pm p$, then the series converges absolutely at the other value as well.

Short answer

Therefore, if the original series absolutely converges at one end of the interval of convergence, it also absolutely converges at the other end.

Step-by-step solution

Step 1. Given information.

The power series is$\sum _{i=0}^{\infty }{a}_k{x}^k$

Calculation

Let us consider the power series $\sum _{i=0}^{\infty }{a}_k{x}^k$with radius of convergence$\rho$

At $\rho$the series would be

$\sum _{k=0}^{\infty }{\left|a_k{x}^k\right|}_{k=\rho }=\sum _{k=0}^{\infty }\left|a_k{\rho }^k\right|$

When the original series is assessed at $\rho$, this is exactly the series of absolute values.

Therefore, if the original series absolutely converges at one end of the interval of convergence, this also absolutely converges at the other end.