Chapter 9.5, Problem 24E

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361

Chapter
Section

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361
Textbook Problem

# Determining Convergence or DivergenceIn Exercises 9-30, determine the convergence or divergence of the series. ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) !

To determine

Whether the series n=0(1)n(2n+1)! is convergent or divergent.

Explanation

Given: The series n=0(1)n(2n+1)!.

Formula used: The Alternating Series test

Let an>0. The alternating series

n=1(1)nan and n=1(1)n+1an converges when the following two conditions hold good.

1. limnan=0, and

2. an+1<an,n

Absolute and conditional convergence test explain that

3. The series n=1an is absolutely convergent when the series n=1|an| converges.

4. The series n=1an is conditionally convergent when the series n=1an converges but the series n=1|an| diverges.

Calculation: Here, let us consider the series n=0(1)n(2n+1)!. Since the series has both positive and negative elements, it can be concluded that the series is an alternating series.

Here an=1(2n+1)!. Since an0 for all positive n and an goes to zero as n goes to , so the first condition of the alternating series test is held true.

Now,

1(2n+2)!<1(2n+1)!,n>0an+1<an,n>0

Now, the series is monotonically decreasing for all n greater than 1, hence it also holds the second criteria of the alternating series test

24)

To determine

Whether given series convergent or divergent.

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