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#

Related questions
See all questions in Integral Test for Convergence of an Infinite Series
Impact of this question
6917 views around the world
You can reuse this answer
Creative Commons License