Q. 63 - Calculus

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.

Step-by-Step Solution

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 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 ∞   a k x − x 0 k 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.