Q. 19 - Calculus

Question

Let $a_k\ne 0$for each value of $k$, and let $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$ be a power series in $x$ with a positive and finite radius of convergence $p$. What is the radius of convergence of the power series $\sum _{k=0}^{\mathrm{\infty }} \frac{1}{a_k}{x}^k$ ?

Step-by-Step Solution

Answer

Ans: The radius of convergence of the power series  $\sum _{k=0}^{\mathrm{\infty }} \frac{1}{a_k}{x}^k$ is $\frac{1}{p}$.

Step 1. Given information.

given,

    $\sum _{k=0}^{\mathrm{\infty }} \frac{1}{a_k}{x}^k$

Step 2. Solution:

 So, we try to evaluate the constant term in the power series using the radius of convergence.

Therefore, let us consider $b_k=a_k{x}^k so b_{k+1}=a_{k+1}{x}^{k+1}$

Now apply the ratio test for absolute convergence, that is 

     $\underset{k\to \mathrm{\infty }}{lim} \left|\frac{b_{k+1}}{b_k}\right|=\underset{k\to \mathrm{\infty }}{lim} \left|\frac{a_{k+1}}{a_k}x\right|$

So according to the ratio test for absolute convergence, the series will converge only when $\left|\frac{a_{k+1}}{a_k}x\right|<1$

Implies that 

  $\left|x\right|=\frac{a_k}{a_{k+1}}$

  Where, $\left|x\right|=\frac{a_k}{a_{k+1}}$  is the radius of convergence of the power series $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$.

   Since we have already considered the radius of convergence of the power series  $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$ is $p$.

Therefore,  $\frac{a_k}{a_k+1}=p$

Step 3. Now, to find the radius of convergence of the power series &#8721; k = 0 &#8734; &#8202; 1 a k x k

  So, here

  $\begin{array}{r}\underset{k\to \mathrm{\infty }}{lim} \left|\frac{b_{k+1}}{b_k}\right|=\underset{k\to \mathrm{\infty }}{lim} \left|\frac{\frac{1}{a_{k+1}}{x}^{k+1}}{\frac{1}{a_k}{x}^k}\right|\\ =\underset{k\to \mathrm{\infty }}{lim} \left|\frac{a_k}{a_{k+1}}x\right|\end{array}$

Since  $\frac{a_k}{a_{k+1}}=p$, therefore

  $\underset{k\to \mathrm{\infty }}{lim} \left|\frac{b_{k+1}}{b_k}\right|=\underset{k\to \mathrm{\infty }}{lim} \left|\rho x\right|$

So according to the ratio test for absolute convergence, the series will converge only when  |px|<1

That is,

  $\left|x\right|<\frac{1}{p}$

Step 4. Hence,

The radius of convergence of the power series $\sum _{k=0}^{\mathrm{\infty }} \frac{1}{a_k}{x}^k$ is $\frac{1}{p}$.