Question

Tính nhanh C=\(\frac{3}{(1\cdot 2)^{2}}+\frac{5}{(2\cdot 3)^{2}}+\frac{7}{(3\cdot 5)^{2}}+...+\frac{2n+1}{[n\cdot (n+1)^{2}]}\)

Giúp mình nhanh nhé tối mình phải nộp rồi! Cảm ơn các bạn nhiều!

Answer 1

\(\frac{2n+1}{\left[n\left(n+1\right)\right]^2}=\frac{\left(n+1\right)^2-n^2}{n^2\left(n+1\right)^2}=\frac{1}{n^2}-\frac{1}{\left(n+1\right)^2}\)

\(\Rightarrow C=\frac{1}{1}-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+...+\frac{1}{n^2}-\frac{1}{\left(n+1\right)^2}\)

\(\Rightarrow C=1-\frac{1}{\left(n+1\right)^2}\)