To determine The first four terms and four partial sums of the series ∑ n = 1 ∞ ( 1 n − 1 n + 1 ) and check whether the series appear to be convergent or not. Explanation Consider the infinite series ∑ n = 1 ∞ ( 1 n − 1 n + 1 ) . Obtain the first four terms of the sequence by substituting n = 1 , 2 , 3 and 4 in the expression 1 n − 1 n + 1 . a 1 = 1 1 − 1 1 + 1 = 1 − 1 2 = 1 2 Similarly, when n = 2 the sequence becomes, a 2 = 1 2 − 1 2 + 1 = 1 2 − 1 3 = 3 − 2 6 = 1 6 When n = 3 the sequence becomes, a 3 = 1 3 − 1 3 + 1 = 1 3 − 1 4 = 4 − 3 12 = 1 12 Further, when n = 4 the sequence becomes, a 4 = 1 4 − 1 4 + 1 = 1 4 − 1 5 = 5 − 4 20 = 1 20 Therefore, the first four terms a 1 , a 2 , a 3 and a 4 of the given sequence are 1 2 , 1 6 , 1 12 and 1 20 _ , respectively. Let the partial sum S n be the sum of first n terms of the series, that is, S n = ∑ k = 1 n ( 1 k − 1 k + 1 ) . Simply the above expression as follows. S n = ∑ k = 1 n ( 1 k − 1 k + 1 ) = ∑ k = 1 n ( k + 1 − k ( k ) ( k + 1 ) ) = ∑ k = 1 n ( 1 ( k ) ( k + 1 ) ) Obtain the first four partial sums of the above series as follows

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

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ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 30.1, Problem 42E

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The first four terms and four partial sums of the series ∑n=1∞(1n−1n+1) and check whether the series appear to be convergent or not.

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Chapter 30.1, Problem 41E

Chapter 30.1, Problem 43E

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Chapter 30 Solutions

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

Ch. 30.1 - Prob. 1PE Ch. 30.1 - Prob. 2PE Ch. 30.1 - Prob. 1E Ch. 30.1 - Prob. 2E Ch. 30.1 - Prob. 3E Ch. 30.1 - Prob. 4E Ch. 30.1 - Prob. 5E Ch. 30.1 - Prob. 6E Ch. 30.1 - Prob. 7E Ch. 30.1 - Prob. 8E

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