   Chapter 11.2, Problem 40E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent. If it is convergent, find its sum.40. ∑ n = 1 ∞ ( 3 5 n + 2 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=1(35n+2n).

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=1(35n+2n)=n=135n+n=12n=3n=115n+2n=11n

=3n=1(15)n+2n=11n (1)

Here, n=11n is harmonic series and it is divergent.

Note that, a nonzero multiple of a divergent series is also divergent.

Therefore, the series 2n=11n is divergent

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