Q. 70 - Calculus

Question

$$\begin{gathered} Let \sum _{k=0}^{\infty }{a}_k{x}^k be a power series with a positive and \\ finite radius of convergence \rho , and let c\ne 0. Prove that \\ the series \sum _{k=0}^{\infty }{a}_k{\left(cx\right)}^k has radius of convergence \frac{\rho }{\left|c\right|}. \end{gathered}$$

Step-by-Step Solution

Answer

The radius of convergence for series$\sum _{k=0}^{\infty }{a}_k{\left(cx\right)}^k$ is$\frac{\rho }{\left|c\right|}$

Step. 1 Finding the value of ρ for the series ∑ k = 0 ∞ a k x k

Ratio Test for absolute convergence for series $\sum _{k=0}^{\infty }{a}_k{x}^k$:

$$\begin{gathered} \underset{k\to \infty }{\lim }\left|\frac{a_{k+1} x^{k+1}}{a_K x^k}\right|=\underset{k\to \infty }{\lim }\left|\frac{a_{k+1} x^{k}x}{a_K x^k}\right| \\ =\left|\frac{a_{k+1}}{a_K} x\right| ............( Evaluating the Limits) \end{gathered}$$

As series $\sum _{k=0}^{\infty }{a}_k{x}^k$ converges only when,$\left|x\frac{a_{k+1}}{a_K}\right|<1$

and therefore  

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

So, the Radius of convergence = $\rho$
 = $\frac{a_k}{a_{k+1}}$.

Step. 2 Finding Radius of Convergence for the Series &#8721; k = 0 &#8734; a k ( c x ) k

Ratio Test for absolute convergence for series $\sum _{k=0}^{\infty }{a}_k{\left(cx\right)}^k$:

$$\begin{gathered} \underset{k\to \infty }{\lim }\left|\frac{a_{k+1} {(c x)}^{k+1}}{a_K {(c x)}^k}\right|=\underset{k\to \infty }{\lim }\left|\frac{a_{k+1} {(c x)}^k(c x)}{a_K {(c x)}^k}\right| \\ =\left|\frac{a_{k+1}}{a_K} c x\right| ............( Evaluating the Limits) \end{gathered}$$

As series $\sum _{k=0}^{\infty }{a}_k{\left(cx\right)}^k$ converges only when,

$$\begin{gathered} \left|\frac{a_{k+1}}{a_K} c x\right|<1 \\ \left|x\right|<\frac{\frac{a_{k+1}}{a_K}}{\left|c\right|} \\ \left|x\right|<\frac{\rho }{\left|c\right|} \end{gathered}$$

and

therefore  

Radius of convergence = $\frac{\rho }{c}$ .

Hence Proved.