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Question
Math
Posted 7 months ago

6. Which of the following statements about the series n=1(1)n1+n\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
Consider the series n=1(1)n1+n\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}. We need to determine its convergence properties
step 2
To check for absolute convergence, we examine the series n=1(1)n1+n=n=111+n\sum_{n=1}^{\infty} \left| \frac{(-1)^{n}}{1+\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{1}{1+\sqrt{n}}
step 3
Note that 11+n\frac{1}{1+\sqrt{n}} is a positive, decreasing function. We compare it to the integral 111+xdx\int_{1}^{\infty} \frac{1}{1+\sqrt{x}} \, dx
step 4
Evaluate the integral 111+xdx\int_{1}^{\infty} \frac{1}{1+\sqrt{x}} \, dx. Let u=xu = \sqrt{x}, then du=12xdxdu = \frac{1}{2\sqrt{x}} \, dx and dx=2ududx = 2u \, du
step 5
The integral becomes 12udu1+u\int_{1}^{\infty} \frac{2u \, du}{1+u}. This integral diverges because the integrand behaves like 2uu=2\frac{2u}{u} = 2 for large uu
step 6
Since the integral diverges, the series n=111+n\sum_{n=1}^{\infty} \frac{1}{1+\sqrt{n}} does not converge absolutely
step 7
Next, we check for conditional convergence using the Alternating Series Test. The series n=1(1)n1+n\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}} is an alternating series
step 8
For the Alternating Series Test, we need to check if an=11+na_n = \frac{1}{1+\sqrt{n}} is decreasing and limnan=0\lim_{n \to \infty} a_n = 0
step 9
The function an=11+na_n = \frac{1}{1+\sqrt{n}} is decreasing because the denominator 1+n1+\sqrt{n} is increasing. Also, limn11+n=0\lim_{n \to \infty} \frac{1}{1+\sqrt{n}} = 0
step 10
Since both conditions of the Alternating Series Test are satisfied, the series n=1(1)n1+n\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}} converges conditionally
Answer
The series converges conditionally.
Key Concept
Alternating Series Test
Explanation
The series n=1(1)n1+n\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}} converges conditionally because it satisfies the conditions of the Alternating Series Test: the terms decrease in magnitude and approach zero.

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