48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) 3ᵏ / (k² + 1)

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{3^k}{k^2 + 1} \). This is an infinite series with terms involving an exponential numerator and a polynomial denominator.

Consider the behavior of the terms \( a_k = \frac{3^k}{k^2 + 1} \) as \( k \to \infty \). Since \( 3^k \) grows exponentially and \( k^2 + 1 \) grows polynomially, the numerator grows much faster than the denominator.

Recall the Divergence Test (also called the nth-term test for divergence): if \( \lim_{k \to \infty} a_k \neq 0 \), then the series diverges. Calculate \( \lim_{k \to \infty} \frac{3^k}{k^2 + 1} \).

Since \( 3^k \) grows without bound and \( k^2 + 1 \) grows much slower, the limit of \( a_k \) as \( k \to \infty \) is infinite, which is not zero.

Conclude that by the Divergence Test, the series \( \sum_{k=1}^{\infty} \frac{3^k}{k^2 + 1} \) diverges.

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Geometric Series and Exponential Growth

A geometric series has terms with a constant ratio between consecutive terms, often involving exponential expressions like 3^k. Understanding how exponential growth compares to polynomial growth (like k²) is crucial for analyzing the behavior of series terms as k approaches infinity.

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Geometric Series

Divergence Test (Test for Divergence)

This test states that if the limit of the terms of a series does not approach zero, the series diverges. It is a quick initial check to determine if further tests are necessary, especially useful when terms involve exponential functions.

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Divergence Test (nth Term Test)

Comparison Test and Limit Comparison Test

These tests compare the given series to a known benchmark series to determine convergence or divergence. By comparing 3^k/(k²+1) to a geometric series like 3^k, one can conclude about the behavior of the original series based on the known properties of the comparison series.

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