Question

Let $\sum _{k=0}^{\mathrm{\infty }} a_k{\left(x-x_0\right)}^k$be a power series in $x-x_0$with a finite radius of convergence $p$. Prove that if the series converges absolutely at $x_0\pm p$, then the series converges absolutely at the other value as well.

Short answer

Ans: If the original series converges absolutely at one endpoint of the interval of convergence, it converges absolutely at the other endpoint as well.

Step-by-step solution

Step 1. Given information.

given,

$\sum _{k=0}^{\mathrm{\infty }} a_k{\left(x-x_0\right)}^k$

Step 2. Consider the power series  ∑k=0∞ akx−x0k with the radius of convergence p.

At $x_0+p$consider the series.

$\sum _{k=0}^{\mathrm{\infty }} \left|a_k{\left((x+\rho )-x_0\right)}^k\right|=\sum _{k=0}^{\mathrm{\infty }} \left|a_k{\rho }^k\right|$

This is precisely the series of absolute values when the original series is evaluated at $x_0-p$

Step 3. Thus,

Therefore, if the original series converges absolutely at one endpoint of the interval of convergence, it converges absolutely at the other endpoint as well.