Q 68. - Calculus

Question

Find the values of x for which the series $\sum _{k=0}^{\infty }{\left(\frac{3}{x}\right)}^k$ converges.

Step-by-Step Solution

Answer

The series $\sum _{k=0}^{\infty }{\left(\frac{3}{x}\right)}^k$ converges only for $x>3$ or $x<-3$.

Step 1. Given information.

Given a series $\sum _{k=0}^{\infty }{\left(\frac{3}{x}\right)}^k$.

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  $\sum _{k=0}^{\infty }{\left(\frac{3}{x}\right)}^k$ has $r=\frac{3}{x}$
.

For the series to converge, $\left|\frac{3}{x}\right|<1$.

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

It follows that $\sum _{k=0}^{\infty }{\left(\frac{3}{x}\right)}^k$ converges for all $x>3$ or $x<-3$.