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Today's calculation of Integral 864

Source: 2013 Nippon Medical School entrance exam

January 30, 2013
calculusintegrationinequalitieslimitcalculus computations

Problem Statement

Let m, nm,\ n be positive integer such that 2≤m<n2\leq m<n.
(1) Prove the inequality as follows.
n+1−mm(n+1)<1m2+1(m+1)2+⋯+1(n−1)2+1n2<n+1−mn(m−1)\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}
(2) Prove the inequality as follows.
32≤lim⁡n→∞(1+122+⋯+1n2)≤2\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2
(3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows.
2918≤lim⁡n→∞(1+122+⋯+1n2)≤6136\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}
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