How do you use the Integral test on the infinite series sum_(n=1)^oo1/(2n+1)^3 ?

1 Answer
Wataru
Sep 19, 2014

Since the corresponding integral
int_1^infty 1/(2x+1)^3dx converges to 1/36,
the series
sum_{n=1}^infty1/(2n+1)^3 also converges by Integral Test.

Let us evaluate the integral.

int_1^infty 1/(2x+1)^3dx=int_1^infty(2x+1)^{-3}dx

Let u=2x+1. Rightarrow {du}/{dx}=2 Rightarrow dx={du}/2
x: 1 to infty Rightarrow u: 3 to infty

=int_3^infty u^{-3}{du}/2

=1/2lim_{t to infty}int_3^t u^{-3}du

=1/2lim_{t to infty}[u^{-2}/{-2}]_3^t

=-1/4lim_{t to infty}(1/t^2-1/3^2)

=-1/4(0-1/9)

=1/36

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