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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