Question

Math

Posted 5 months ago

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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.
```

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Answer from Sia

Posted 5 months ago

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 $\lim_{n \to \infty} a_n = 0$

step 3

The sequence $a_n = \frac{1}{1+\sqrt{n}}$ is positive, decreasing, and $\lim_{n \to \infty} a_n = 0$

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 $\sum_{n=1}^{\infty} \left|\frac{(-1)^{n}}{1+\sqrt{n}}\right| = \sum_{n=1}^{\infty} \frac{1}{1+\sqrt{n}}$

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.

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