Question

In Exercises \(3-6,\) find the radius of convergence of the power series. $$ \sum_{n=0}^{\infty}(2 x)^{n} $$

Short answer

The radius of convergence of the power series \(\sum_{n=0}^{\infty}(2 x)^{n}\) is 0.5.

Step-by-step solution

Write down the series

First, it's important to write down the series that is to be examined for convergence. The given series is: \(\sum_{n=0}^{\infty}(2 x)^{n}\)

Apply the ratio test

The ratio test can be used to find the radius of convergence of a series. This test involves taking the limit as n approaches infinity of the absolute value of the ratio of the \(n+1\)th term to the \(n\)th term of the series. The absolute value of this limit will give us the radius of convergence. Apply the ratio test for the given series: \(\lim_{{n \to \infty}} \left|\frac{{a_{n+1}}}{{a_n}}\right|\) where \(a_{n+1}\) is the \(n+1\)th term and \(a_n\) is the \(n\)th term of the series.

Simplify the ratio

For the given series, \(a_n = (2 x)^n\), and \(a_{n+1} = (2 x)^{n+1}\). Simplifying the ratio gives: \(\lim_{{n \to \infty}} \left| \frac{{(2 x)^{n+1}}}{{(2 x)^n}} \right|=\left|2x\right|\)

Determine the radius of convergence

The series will converge if the limit found above is less than 1, i.e. \(\left|2x\right|<1\). This simplifies to \(-0.5 < x < 0.5\). Hence, the radius of convergence is 0.5.