Sum of $\frac{\sin n}n$

Through brute force Desmos, one may suspect the following. $\lim\limits_{n\to\infty}2\sum\limits_{k=1}^n\frac{\sin k}k+1=\pi$ . If this is true, is there an intuitive way to understand it?

#Geometry

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

https://math.stackexchange.com/questions/13490/proving-that-the-sequence-f-nx-sum-limits-k-1n-frac-sinkxk-is

Patrick Corn - 1 year, 2 months ago

i think it can be pretty easily explained through corresponding Fourier series

Nick Kent - 1 year, 2 months ago

This might help you

Zakir Husain - 1 year ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Sum of sin⁡nn\frac{\sin n}nnsinn​

Comments

Sum of $\frac{\sin n}n$