Question

Math

Posted 7 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 7 months ago

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.

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