Q. 65 - Calculus

Question

Prove that if $\left(x_0-\rho ,x_0+\rho \right]$ is the interval of convergence for the series $\sum _{k=0}^{\mathrm{\infty }} a_k{\left(x-x_0\right)}^k$, then the series converges conditionally at $x_0+p$. 

Step-by-Step Solution

Answer

Ans:  Therefore, the series converges conditionally at $x_0+p$

Step 1. Given formation.

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 and x 0 − ρ , x 0 + ρ is the interval of convergence of the series.

  At $x_0+p$, the series would become

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

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

Therefore, the series converges conditionally at $x_0+p$.