Q. 17 - Calculus

Question

Let $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$ be a power series in $x$ with a positive and finite radius of convergence $p$. Explain why the ratio test for absolute convergence will fail to determine the convergence of this power series when $x=p$ or when $x=-p$.

Step-by-Step Solution

Answer

Ans:  The ratio test for absolute convergence fails at the endpoints of the interval of convergence.

Step 1. Given information.

given,

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

Step 2. Solution.

 If $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$is a power series in $x$ with a positive and finite radius of convergence $p$ then the limit $\underset{k\to \mathrm{\infty }}{lim} \left|\frac{b_{k+1}}{b_k}\right|=\underset{k\to \mathrm{\infty }}{lim} \left|\frac{a_{k+1}}{a_k}\rho \right|$ which is equal to $1$.

Step 3. Thus,

The ratio test for absolute convergence fails at the endpoints of the interval of convergence.