Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)5ᵏ / 2²ᵏ⁺¹
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14. Sequences & Series
Convergence Tests
2:22 minutesProblem 10.R.65
Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k√k / k³
Verified step by step guidance1
First, rewrite the general term of the series to simplify it. The term is given as \(\frac{k \sqrt{k}}{k^3}\). Recall that \(\sqrt{k} = k^{1/2}\), so rewrite the term as \(\frac{k \cdot k^{1/2}}{k^3}\).
Combine the powers of \(k\) in the numerator: \(k \cdot k^{1/2} = k^{1 + 1/2} = k^{3/2}\). So the term becomes \(\frac{k^{3/2}}{k^3}\).
Simplify the fraction by subtracting exponents in the denominator from the numerator: \(k^{3/2 - 3} = k^{-3/2}\).
Now, the series can be written as \(\sum_{k=1}^\infty k^{-3/2}\). Recognize this as a p-series of the form \(\sum_{k=1}^\infty \frac{1}{k^p}\) where \(p = \frac{3}{2}\).
Apply the p-series convergence test: a p-series converges if and only if \(p > 1\). Since \(\frac{3}{2} > 1\), conclude that the series converges.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of series like the one given.
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Convergence Tests
Convergence tests are methods used to determine if an infinite series converges or diverges. Common tests include the Comparison Test, Ratio Test, Root Test, and p-Series Test. Choosing an appropriate test depends on the form of the series terms.
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Simplifying Series Terms
Before applying convergence tests, it is important to simplify the general term of the series. For example, rewriting k√k / k³ as k^(1 + 1/2) / k³ = k^(3/2) / k³ = k^(-3/2) helps identify the type of series and apply the correct convergence test.
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