Textbook Question
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.
2 / 4² + 2 / 5² + 2 / 6² + ⋯
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14. Sequences & Series
Convergence Tests
1:30 minutesProblem 10.4.60
Textbook Question
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)
Verified step by step guidance1
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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Video duration:
1mKey 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.
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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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05:44Divergence 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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07:45Limit Comparison Test
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