Question

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.∑ (from k = 1 to ∞) (√k / k − 1)²ᵏ

Accepted Answer

First, rewrite the general term of the series to understand its structure clearly. The term is given by $$a_k = \frac{\sqrt{k}}{k} - 1$$^{2k}$$. Since the expression is ambiguous, clarify the term as a_k$$ = \left( \frac{\sqrt{k}}{k} - 1 \$$right)^{2k} or$$ $$a_k = \frac{\sqrt{k}}{k - 1$$^{2k}$$}. $$Assuming the problem means $$a_k = \left( \frac{\sqrt{k}}{k} - 1 \$$right)^{2k}$$, simplify the base inside the parentheses first. Simplify the base inside the parentheses: $$\frac{\sqrt{k}}{k} = \frac{$$k^{1/2}$$}{k} = $$k^{-1/2}. So $$the base becomes $$k^{-1/2} - 1$. Rewrite the term as a_k$$ = \left( $$k^{-1/2} - 1$$ \$$right)^{2k}. $$Since the term is raised to the power $$2k$$, consider using the Root Test or the Ratio Test to determine convergence. The Root Test is often simpler for terms raised to the power $$k. $$The Root Test involves computing $$\lim_{k 	o \infty} \sqrt[2k]{|a_k|}. $$Calculate the $$2k-th $$root of $$|a_k|$$: $$\sqrt[2k]{|a_k|} = |k$$^{-1/2}$$ - 1|. $$Then, find the limit as $$k$$ approaches infinity of this expression. This limit will help determine the behavior of the series. Interpret the limit from the Root Test: if the limit is less than 1, the series converges absolutely; if it equals 1, the test is inconclusive; if greater than 1, the series diverges. Use this conclusion to classify the series as absolutely convergent, conditionally convergent, or divergent.

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (√k / k − 1)²ᵏ

Verified video answer for a similar problem:

Key Concepts

Absolute and Conditional Convergence

Root Test

Behavior of Terms Involving Powers and Radicals