   Chapter 11.2, Problem 47E

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.47. ∑ n = 1 ∞ ( e 1 / n − e 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=1(e1ne1(n+1)).

Here, an=e1ne1(n+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=1(e1ne1(n+1)). Then,

sn=k=1ne1ke1(k+1)=(e11e1(1+1))+(e12e1(2+1))+(e13e1(3+1))++(e1n1e1(n1+1))+(e1ne1(n+1<

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