Q. 10 - Calculus

Question

Let $\sum _{k=1}^{\mathrm{\infty }} {\mathrm{Cr}}^k$ be a series with c and $r$$\in \mathrm{ℝ}$ . Explain why the convergence of this series depends only upon the magnitude of r and not c .

Step-by-Step Solution

Answer

The geometric behaviour is independent of c . The ratio is dependent on r . Therefore the convergence of the series is governed by the ratio r only .

Step 1. Given information

We have been given a series to find out the parameters on which its convergence depends 

Step 2. Explain why the convergence depends upon r and not on c .

Consider the geometric series $\sum _{k=1}^{\mathrm{\infty }} c{r}^k$

Our objective is to prove  that convergence depends on r only and not on c .

Step 3. Expansion of the series

$\sum _{k=1}^{\mathrm{\infty }} c{r}^k=cr+c{r}^2+c{r}^3+\dots$ (Series in expanded form )

$=c\left(r+r^2+r^3+\dots \right)$ (Factorize)

$=c\sum _{k=1}^{\mathrm{\infty }} r^k$

Thus , if c is any real number then $\sum _{k=1}^{\mathrm{\infty }} {\mathrm{Cr}}^k=C\sum _{k=1}^{\mathrm{\infty }} r^{\ast }$ holds .

Therefore the convergence depends only on r .