Question

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} $$

Short answer

By using the ratio test, it is determined that the series \( \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} \) diverges.

Step-by-step solution

Definition of the Ratio Test

The Ratio Test states that for a series \( \sum_{n=1}^{\infty} a_{n} \), the limit L = \( \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_{n}} \right| \), if L < 1 then the series absolutely converges, if L > 1 then series diverges, and if L = 1 the test is inconclusive.

Apply the Ratio Test

The Ratio Test is applied to the given series \( \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} \) by taking the ratio \( _{n+1}/a_{n} \), giving \( \frac{\frac{2^{n+1}}{(n+1)^{2}}}{\frac{2^{n}}{n^{2}}} \). This simplifies to \( \frac{2(n^{2})}{(n+1)^{2}} \) .

Calculate the Limit

Next find the limit as n approaches infinity, \( \lim_{n \rightarrow \infty} \frac{2(n^{2})}{(n+1)^{2}} \). This gives a limit of 2.

Assess the Ratio Test Result

Since the limit from step 3, which is 2, is greater than 1, the Ratio Test determines that the given series diverges.