Question

Q.

Find all values of \(x\) for which the series \(\sum_{k=1}^{∞} \left ( \frac{x}{3} \right )^{k}\) converges.

Short answer

The value of \(x\) lies in the interval \(\left (-3,3 \right )\)

Step-by-step solution

Step 1. Given Information.

The given series \(\sum_{k=1}^{∞} \left ( \frac{x}{3} \right )^{k}\) converges.

To find \(x\) values.

We will do this question by Geometric Series Test which says the series of the type \( \sum a(r)^n\) converges when \(\left| r \right| < 1\).

Here, \(r=\frac{x}{3}\).

For this series to be convergent we must have \(\left| \frac{x}{3}\right|< 1\)

\(\Rightarrow \left|x \right|<3\)

\(\Rightarrow -3<x<3\)

Thus, the series \(\sum_{k=1}^{∞} \left ( \frac{x}{3} \right )^{k}\) is convergent if \(x\epsilon \left ( -3,3 \right )\)