   Chapter 11.2, Problem 46E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent by expressing sn as a telescoping sum (as in Example 8). If it is convergent, find its sum.46. ∑ n = 4 ∞ ( 1 n − 1 n + 1 )

To determine

Whether the series is convergent or divergent and obtain the series sum if the series is convergent.

Explanation

Given:

The series is n=41n1n+1.

Here, an=1n1n+1.

Result used:

If the limit of the partial sums exists and limnsn=L, then the series convergent and its sum is n=1an=L.

Calculation:

Obtain the limit of the partial sums.

Let sn be the nth partial sum of the series n=41n1n+1. Then,

sn=k=4n1k1k+1=(1414+1)+(1515+1)+(1616+1)++(1n1n+1)=(1215)

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