   Chapter 11.2, Problem 38E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent. If it is convergent, find its sum.38. ∑ k = 0 ∞ ( 2 ) − 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=0(2)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=0(2)k=k=01(2)k=k=0(12)k=(12)0+(12)1+(12)2+(12)3+=1+(12)+(12)2+(12)3+

Here, the first term of the series is a = 1 and the common ratio of the series is,

r=121=12

The absolute value of r is,

|r|=|12|=12=0

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