Little Big Sum

Calculus Level 5

When

$S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n n}\right|,$

the value of $S$ is of the form

$\dfrac{1}{2}\sqrt{\log^2(\tan x)+1},$

where $0<x-1<\dfrac{\pi}{4}.$ Find $\lfloor 1000x \rfloor.$

Details and Assumptions

$i$ is the imaginary unit, and not a variable.
$\lfloor \cdots \rfloor$ is the floor function, or greatest integer function.

The answer is 1285.

1 solution

Aman Rajput
May 19, 2021

Using the sum value through calculator $S=\sum_{n=1\to\infty}\frac{\sin n}{i^n n}\approx -0.5+0.6131i$ So we get $|S|=0.791=\frac12\sqrt{\ln^2(\tan x)+1}$ and solving this we get $\tan x \approx 3.4082,0.2934$ Taking inverse tangent within the given domain, $x \approx 1.28539$ So, according to the answer it should be asked $\lfloor 1000(x-1)\rfloor = 285$

So this part of the question seems incorrect to me. Please look into it.

Little Big Sum

The answer is 1285.

1 solution

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