   Chapter 11.2, Problem 35E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent. If it is convergent, find its sum.35. ∑ k = 1 ∞ ( sin 100 ) k

To determine

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

Explanation

Given:

The series is k=1(sin100)k .

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,

k=1(sin100)k=(sin100)1+(sin100)2+(sin100)3+=(sin100)+(sin100)2+(sin100)3+

Here, the first term of the series is a=sin100 and the common ratio of the series is,

r=(sin100)2sin100=sin100

The absolute value of r is,

|r|=|sin100|=|0

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