   Chapter 11.9, Problem 41E

Chapter
Section
Textbook Problem

Use the power series for tan − 1 x to prove the following expression for π as the sum of an infinite series: π = 2 3 ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 3 n

To determine

To prove:

The following expression for π as the sum of an infinite series:

π=23n=0-1n2n+13n

by using the power series for tan-1x

Explanation

1) Concept:

Use the power series for tan-1x and simplify.

2) Calculation:

The power series for tan-1x is

For x<1,

tan-1x=n=0-1nx2n+12n+1

For x=13

tan-113=n=0-1n132n+12n+1

π6=n=0-1n132n+12n+1

Simplify.

=n=0-1n132n+1(2n+1)

=n=0-1n13

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