Proof by pinch theorem lim[n→∞] {1/n^2 + 1/(n+1)^2 +∧+1/(2n)^2} =0

Proof by pinch theorem lim[n→∞] {1/n^2 + 1/(n+1)^2 +∧+1/(2n)^2} =0

In the limit formula, the smallest denominator is n & # 178;, so take all the denominators as N & # 178;, and the whole formula will be enlarged
So there is
0 ≤ 1 / N and 178; + 1 / (n + 1) & 178; +... + 1 / (2n and 178;) ≤ 1 / N and 178; + 1 / N and 178; +... + 1 / N and 178; = (n + 1) / N and 178; = 1 / N and 178; + 1 / N -- > 0, when n -- > ∞
So we can see that the limit of the above two formulas is 0 when n tends to ∞
So the middle limit is 0 when n tends to ∞
Note that when using the pinch criterion to prove, the amount of amplification and reduction should tend to the same limit!