∑(2n+1)/{(n^2)*[(n+1)^2]}=?n是从1开始到无穷大,希望帮我解答

Question

∑(2n+1)/{(n^2)*[(n+1)^2]}=?n是从1开始到无穷大,希望帮我解答

Answer 1

∑(2n+1)/{(n^2)*[(n+1)^2]}
= ∑(1/n^2-1/(n+1)^2)
=1/1^2-1/2^2+1/2^2-1/3^2+.+1/n^2-1/(n+1)^2
=1-1/(n+1)^2
显然,当n趋向于无穷大时,1/(n+1)^2趋向于0
所以,∑(2n+1)/{(n^2)*[(n+1)^2]}=1