Question

$$\begin{gathered} Let\sum _{k=0}^{\infty }{a}_k{x}^{k}beapowerserieswithapositiveand \\ finiteradiusofconvergence\rho ,andletc\ne 0.Provethat \\ theseries\sum _{k=0}^{\infty }{a}_k{\left(cx\right)}^{k}hasradiusofconvergence\frac{\rho }{\left|c\right|}. \end{gathered}$$

Short answer

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

Step-by-step solution

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

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

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

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 ∑k=0∞ak(cx)k

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

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

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

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

and

therefore

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

Hence Proved.