A calculus problem by Sachin Vishwakarma

Calculus Level 4

$\large \displaystyle \lim_{n \to \infty} \sin \left(\frac {n}{ n^2 + 1^2}\right) +\sin \left({\frac {n}{ n^2 + 2^2} }\right)+ \cdots +\sin \left({ \frac {n}{ n^2 + n^2} } \right) = \, ?$

None of these

\pi

\frac{\pi}{4}

\frac{ \pi}{2}

1 solution

Tanishq Varshney
Nov 26, 2015

We know $\sin x = x$ when x approaches zero.

The given series is

$\large{\frac{1}{n} \lim_{n \to \infty} \displaystyle \sum_{r=1}^{n}\frac{1}{1+\frac{r^2}{n^2}}}$

Using reimann integration method

$\large{\displaystyle \int^{1}_{0} \frac{1}{1+x^2} dx}$

Thus the answer is

$\Large{\boxed{\frac{\pi}{4}}}$

What if we divide and multiply by the expression inside sin() and we know that limx->\infty sinx/x=1 so remaining term can be solved by L'Hospital to get the answer as 0

Naman Kapoor - 5 years, 5 months ago

A calculus problem by Sachin Vishwakarma

1 solution

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