40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 1 to ∞) (1 + 2 / k)ᵏ

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Identify the given series: \( \sum_{k=1}^{\infty} \left(1 + \frac{2}{k}\right)^k \). This is an infinite series where each term is \( a_k = \left(1 + \frac{2}{k}\right)^k \).

Recognize that the terms \( a_k \) resemble the form \( \left(1 + \frac{x}{k}\right)^k \), which is related to the expression for \( e^x \) as \( k \to \infty \). Here, \( x = 2 \).

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

Evaluate the limit \( \lim_{k \to \infty} \left(1 + \frac{2}{k}\right)^k = e^2 \), which is a positive finite number greater than zero.

Since the limit of the terms \( a_k \) is not zero, by the Test for Divergence, the series \( \sum_{k=1}^{\infty} \left(1 + \frac{2}{k}\right)^k \) diverges.

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Key Concepts

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

Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the series sums to a finite value.

Root Test

The Root Test evaluates the limit of the k-th root of the absolute value of the terms in a series. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. This test is especially useful for series with terms raised to the k-th power.

Limit Comparison and Growth of Terms

Understanding how the terms (1 + 2/k)^k behave as k approaches infinity is crucial. This expression resembles the form (1 + 1/n)^n, which approaches e, but with a different coefficient. Comparing the limit of terms to known limits helps determine if terms approach zero, a necessary condition for convergence.

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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 = 1 to ∞) k^(–1/5)

Textbook Question

9–15. Geometric sums Evaluate each geometric sum.

{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ

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Textbook Question

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

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Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.