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Calculus Level 3

$\large \lim_{n\to\infty} \frac1n \sum_{k=1}^n \cos\left(\frac{k\pi}{2n}\right)$

If the limit above equals to $A \pi^B$ for integers $A$ and $B$ , find the value of $A\times B$ .

1 solution

Vishwak Srinivasan
Aug 8, 2015

The given limit of summation is a Riemann Integral.

$I = \displaystyle \int_{0}^{1} \cos\left(\dfrac{\pi}{2} x\right) dx$

$t = \dfrac{\pi}{2}x \Rightarrow dt = \dfrac{\pi}{2} dx$

$I = \displaystyle \dfrac{2}{\pi} \int_{0}^{\frac{\pi}{2}} \cos t dt = \dfrac{2}{\pi}$

$A = 2 , B = -1 \Rightarrow A\times B = \boxed{-2}$