Q. 62 - Calculus

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.

Step-by-Step Solution

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 1. Given information.

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

Step 2: 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.