Question

So sánh:

\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+.....+\frac{1}{n^2}\)

Với 1
 

Answer 1

\(\frac{1}{2^2}< \frac{1}{2}.\left(\frac{1}{1}-\frac{1}{3}\right)\)

\(\frac{1}{3^2}< \frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}\right)\)

\(\frac{1}{4^2}< \frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}\right)\)

\(\frac{1}{5^2}< \frac{1}{2}\left(\frac{1}{4}-\frac{1}{6}\right)\)

\(\frac{1}{\left(n-1\right)^2}< \frac{1}{2}\left(\frac{1}{n-2}-\frac{1}{n}\right)\)

\(\frac{1}{n^2}< \frac{1}{2}\left(\frac{1}{n-1}-\frac{1}{n+1}\right)\)

\(A< 1\)