Find limn →∞ (an+1)/(an), and use the ratio test to...

Find $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}},$ and use the ratio test to determine if the series converges or diverges or if the test is inconclusive.
$\sum_{n=1}^{\infty} \frac{3 n+1}{2^{n}}$

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Key Concepts

Ratio Test

The ratio test is a method for determining the convergence or divergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if it is greater than one, the series diverges; and if it equals one, the test is inconclusive.

Limit of the Ratio of Consecutive Terms

Calculating the limit of the ratio a???/a? as n approaches infinity is key to applying the ratio test. This limit indicates the dominant behavior of the series’ terms, especially when they involve combinations of polynomial and exponential functions.

Convergence Criteria

Based on the outcome of the ratio test, the convergence criteria are determined by comparing the computed limit to one. A limit less than one confirms absolute convergence of the series, a limit greater than one indicates divergence, and a limit equal to one means that further investigation is needed as the test does not provide a definitive answer.

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