Textbook Question
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) tan(1 / k)
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14. Sequences & Series
Convergence Tests
2:32 minutesProblem 10.4.31
Textbook Question
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / (k − 2)⁴
Verified step by step guidance1
Identify the given series: \( \sum_{k=3}^{\infty} \frac{1}{(k-2)^4} \). Notice that the term inside the summation can be rewritten as \( \frac{1}{n^4} \) by letting \( n = k - 2 \). This shifts the index to start from \( n=1 \).
Recognize that the series is a p-series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) where \( p = 4 \). The p-series test states that such a series converges if and only if \( p > 1 \).
Since \( p = 4 > 1 \), the p-series test indicates that the series converges.
Optionally, you could apply the Integral Test by considering the function \( f(x) = \frac{1}{x^4} \) for \( x \geq 1 \), which is positive, continuous, and decreasing. Then evaluate the improper integral \( \int_1^{\infty} \frac{1}{x^4} \, dx \) to confirm convergence.
Conclude that by the p-series test (and optionally the Integral Test), the series \( \sum_{k=3}^{\infty} \frac{1}{(k-2)^4} \) converges.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Divergence Test
The Divergence 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 a series cannot converge, but if the limit is zero, the test is inconclusive.
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Integral Test
The Integral Test relates a series to an improper integral by comparing the sum of terms to the integral of a corresponding function. If the integral converges, so does the series; if the integral diverges, the series diverges as well. It requires the function to be positive, continuous, and decreasing.
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p-series Test
A p-series is a series of the form ∑ 1/n^p. It converges if and only if p > 1 and diverges otherwise. This test is useful for quickly determining convergence of series with terms involving powers of n.
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