   Chapter 11.2, Problem 28E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent. If it is convergent, find its sum.28. 1 3 + 2 9 + 1 27 + 2 81 + 1 243 + 2 729 + ⋯

To determine

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

Explanation

Given:

The series is 13+29+127+281+1243+2719+ .

Definition used:

If the sum of all terms of the series is finite, then the series is said to be convergent series.

Result used:

The geometric series n=1arn1 (or) a+ar+ar2+ is convergent if |r|<1 and its sum is a1r , where a is the first term and r is the common ratio of the series.

Calculation:

The given series can be written as follows,

13+29+127+281+1243+2719+=(13+127+1243+)+(29+281+2719+) (1)

Here, both the series 13+127+1243+ and 29+281+2719+ are geometric series.

Obtain the sum of the geometric series 13+127+1243+ if the series is convergent.

Consider 13+127+1243+ .

Here, the first term of the series is, a1=13 and the common ratio of the series is,

r1=12713=12731=19

The absolute value of r1 is,

|r1|=|19|=19=0.1111<1

Since |r1|<1 and by the Result stated above, the geometric series 13+127+1243+ is convergent

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