Q. 64 - Calculus

Question

Prove that if $(-p,p]$ is the interval of convergence for the series $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$, then the series converges conditionally at $p$.

Step-by-Step Solution

Answer

Ans:  Therefore, if the interval of convergence of series is $(-p,p]$ if the original series converges absolutely at one endpoint of the interval of convergence, then the series converges conditionally at $x=p$.

Step 1. Given information.

given,

     $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$

Step 2. Consider the power series ∑ k = 0 ∞   a k x k with a radius of convergence p

  At $p$ the series would be

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

This is precisely the series of absolute values when the original series is evaluated at $p$ which converges conditionally.

Step 3. Thus,

Therefore, if the interval of convergence of series is $(p,p ]$ if the original series converges absolutely at one endpoint of the interval of convergence, then the series converges conditionally at $x=p$.