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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

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)