   Chapter 11.2, Problem 40E

Chapter
Section
Textbook Problem

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

To determine

The series is convergent or divergent and if it is convergent, find its sum

Explanation

1) Concept:

Theorem: If n=1an & n=1bn  are convergent series, then the series

n=1an+bn, n=1(an-bn)   &

n=1(an+bn)=n=1an+n=1bn

n=1(an-bn)=n=1an-n=1bn

2) Given:

n=135n+2n

3) Calculation:

Consider the given series.

n=135n+2n

By using the concept,

n=135n+2n=n=135n+n=12n

First, check the convergence of n=135n+2n

Assume that  n=135n+2n  is convergent.

Consider n=135n

n=135n=35n=115n-1

35n=115n-1 is a geometric series

Here,a=35, r=15

By using the concept,

r=15=0.2<1

By using the test of convergence of series,n=135n   is convergent.

Now, consider n=12n

n=12n=2n=11n

Check the convergence of the series n=11n

For the particular series n=11n,  consider the partial sum of s2,s4,,s8,s16,s32,

s2=1+12

s4=1+12+13+14

s8=1+12+13+14+15</

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