Question

Q.

A series of monomials: Find all values of \(x\) for which the series \( \sum_{k=1}^{∞} (4x)^k\) converges.

Short answer

The value of \(x\) lies in the interval \(\left (\frac{-1}{4},\frac{1}{4} \right )\)

Step-by-step solution

Step 1. Given information

The given series \( \sum_{k=1}^{∞} (4x)^k\) converges.

Step 2. Finding the \(x\) value.

For a series of type \( \sum a(r)^n\) to converge we must have that \(\left| r \right| < 1\).

Here, \(r=4x\)

For this series to be convergent we must have \(\left|4x \right| < 1\)

\(\Rightarrow -1<4x<1\)

\(\Rightarrow \frac{-1}{4}<x<\frac{1}{4}\)

Thus, the series \( \sum_{k=1}^{∞} (4x)^k\) is convergent if \(x \epsilon \left (\frac{-1}{4},\frac{1}{4} \right )\)