   Chapter 11.2, Problem 34E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent. If it is convergent, find its sum.34. ∑ n = 1 ∞ 2 n + 4 n e n

To determine

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

Explanation

Given:

The series is n=12n+4nen .

Result used:

(1) The geometric series n=1arn1 (or) a+ar+ar2+ is convergent if |r|<1 and its sum is a1r .

(2) The geometric series n=1arn1 (or) a+ar+ar2+ is divergent if |r|1 , where a is the first term and r is the common ratio of the series.

Calculation:

The given series can be written as follows,

n=12n+4nen=n=1[2nen+4nen]=n=1[(2e)n+(4e)n]

n=12n+4nen=n=1(2e)n+n=1(4e)n (1)

Here, both the series n=1(2e)n and n=1(4e)n are geometric series.

Consider the series, n=1(2e)n=(2e)+(2e)2+(2e)3+ .

Here, the first term of the series is, a1=2e and the common ratio of the series is,

r1=(2e)22e=2e

The absolute value of r1 is,

|r1|=|2e|=2e=0

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