6. Which of the following statements about the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$ is true?
(A) The series converges absolutely.
(B) The series converges conditionally.
(C) The series converges but neither conditionally nor absolutely.
(D) The series diverges.

Question

6. Which of the following statements about the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$ is true?
(A) The series converges absolutely.
(B) The series converges conditionally.
(C) The series converges but neither conditionally nor absolutely.
(D) The series diverges.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the convergence of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$, we consider the alternating series test. ‖ ‖ step 2 ⋮ The alternating series test states that an alternating series $\sum_{n=1}^{\infty} (-1)^{n}a_n$ converges if the sequence $a_n$ is decreasing and $\lim_{n o \infty} a_n = 0$. ‖ ‖ step 3 ⋮ The sequence $a_n = \frac{1}{1+\sqrt{n}}$ is positive, decreasing, and $\lim_{n o \infty} a_n = 0$. ‖ ‖ step 4 ⋮ Therefore, the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$ converges conditionally by the alternating series test. ‖ ‖ step 5 ⋮ To check for absolute convergence, we consider the series $\sum_{n=1}^{\infty} \left|\frac{(-1)^{n}}{1+\sqrt{n}} ight| = \sum_{n=1}^{\infty} \frac{1}{1+\sqrt{n}}$. ‖ ‖ step 6 ⋮ The series $\sum_{n=1}^{\infty} \frac{1}{1+\sqrt{n}}$ is a p-series with $p = \frac{1}{2}$, which diverges. ‖ ‖ step 7 ⋮ Since the series does not converge absolutely, it converges conditionally. ‖ ∻Answer∻ ⚹ (B) The series converges conditionally. ⚹ ∻Key Concept∻ ⚹Alternating Series Test and Absolute Convergence⚹ ∻Explanation∻ ⚹The series converges conditionally because it satisfies the alternating series test, but it does not converge absolutely since the absolute series diverges.⚹◊What is the alternating series test and how can it be applied to determine the convergence of the given series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$?⍭ Generate me a similar question◊

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