无穷数列{1/（4n^2-1）}的各项和等于多少?

Question

无穷数列 {1/（4n^2-1）}的各项和等于多少?

Answer 1

an=1/（4n^2-1)=1/(2n-1)(2n+1)=1/(4n-2)-1/(4n+2)
a(n-1)=1/(4n-6)-1/(4n-2)
an+a(n-1)+.+a1=1/(4n-2)-1/(4n+2)+1/(4n-6)-1/(4n-2)+.+1/2-1/6=(1/2)-1/(4n+2)