Question

Cho \(a_k\)= \(\frac{2k+1}{k^2\left(k+1\right)^2}\) với k >= 1  chứng minh S = a1 +a2 +a3+.....+a 50 >1/3

Answer 1

ae ơi đề bài lại như này nhé chứng minh a 1 + a2 +....+a99 <1

Answer 2

\(a_k=\frac{2k+1}{k^2\left(k+1\right)^2}=\frac{k^2+2k+1-k^2}{k^2\left(k+1\right)^2}=\frac{\left(k+1\right)^2}{k^2\left(k+1\right)^2}-\frac{k^2}{k^2\left(k+1\right)^2}=\frac{1}{k^2}-\frac{1}{\left(k+1\right)^2}\)

\(S=\frac{1}{1^2}-\frac{1}{\left(1+1\right)^2}+\frac{1}{2^2}-\frac{1}{\left(2+1\right)^2}+\frac{1}{3^2}-\frac{1}{\left(3+1\right)^2}+...+\frac{1}{99^2}-\frac{1}{\left(99+1\right)^2}\)

\(S=1-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{3^2}-\frac{1}{4^2}+...+\frac{1}{99^2}-\frac{1}{100^2}=1-\frac{1}{100^2}< 1\) ( đpcm ) 

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