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 nature of the convergence of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$, we refer to the asksia-ll calculator result. ‖ ‖ step 2 ⋮ According to the asksia-ll calculator, the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$ converges to approximately $-0.278283$. ‖ ‖ step 3 ⋮ The asksia-ll calculator also confirms that the series converges. ‖ ‖ step 4 ⋮ To check for absolute convergence, we consider the series $\sum_{n=1}^{\infty} \left|\frac{(-1)^{n}}{1+\sqrt{n}}\right|$. ‖ ‖ step 5 ⋮ Since $\left|\frac{(-1)^{n}}{1+\sqrt{n}}\right| = \frac{1}{1+\sqrt{n}}$ and $\frac{1}{1+\sqrt{n}} \geq \frac{1}{1+n}$, we compare it to the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$, which is known to diverge. ‖ ‖ step 6 ⋮ Because the terms $\frac{1}{1+\sqrt{n}}$ do not decrease to zero fast enough, the series does not converge absolutely. ‖ ∻Answer∻ ⚹ (B) The series converges conditionally. ⚹ ∻Key Concept∻ ⚹Conditional Convergence⚹ ∻Explanation∻ ⚹The series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$ converges, but not absolutely, because the absolute value of the series does not converge. This is known as conditional convergence.⚹◊What is the alternating series test and how can it be applied to the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$?⍭ Generate me a similar question◊

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