Q 67. - Calculus

Question

Find the values of x for which the series $\sum _{k=0}^{\infty }{x}^k$ converges.

Step-by-Step Solution

Answer

The series $\sum _{k=0}^{\infty }{x}^k$ converges only for $-1<x<1$.

Step 1. Given information.

Given a series $\sum _{k=0}^{\infty }{x}^k$.

Step 2. Find all values of x for which the series converges.

A geometric series is of the form $\sum _{k=0}^{\infty }c{r}^k$ for some constants c and r.

Suppose r is a non-zero real number, then $\sum _{k=0}^{\infty }c{r}^k$ converges to $\frac{c}{1-r}$ if and only if $\left|r\right|<1$.

Here, the series $\sum _{k=0}^{\infty }{x}^k$ has $r=x$.

For the series to converge, $\left|x\right|<1$.

Note that if $\left|y\right|<a$, it follows that $-a<y<a$.

It follows that $\sum _{k=0}^{\infty }{x}^k$ converges for all $-1<x<1$.