Reciprocal Hyperbolic Sine Summation

Find a closed form of

$\sum_{n=1}^{\infty} \dfrac{1}{\sinh 2^{n} }$

Notation: $\sinh$ denotes hyperbolic sine function.

This is a part of the set Formidable Series and Integrals

#Calculus

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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`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Ishan Singh The answer is 2/(e^2 -1)

Aaghaz Mahajan - 2 years, 8 months ago

@Ishan Singh Simply convert this into a double summation after a few manipulations...........And Use the property that every natural number can be expressed as (2^n)*(2k+1) for n and k belonging to Whole numbers........

Aaghaz Mahajan - 2 years, 8 months ago

Nice. I just telescoped the summation.

Ishan Singh - 2 years, 8 months ago

Well that is cool as well!!! But I am much more comfortable with Double sums...... :)
Hey, can I ask you something??? I have followed you for a long time on Brilliant.......could you suggest me where to learn all those amazing integration/Binomial tricks you have up your sleeve??? Like I know stuff, but usually my manipulations turn out to be tedious that it consumes two pages or so..........any tips???

Aaghaz Mahajan - 2 years, 8 months ago

@Aaghaz Mahajan – I suggest starting with the book Inside Interesting Integrals and then moving on to advanced books like Limits, Series, and Fractional Part Integrals and Problems and Theorems in Analysis I.

For Combinatorial stuff (binomial sums, generating functions, discreet calculus etc.) you can try the books Concrete Mathematics and Generatingfunctionology.

Try reading up Wikipedia/Mathworld pages on topics and the associated references/papers from Google Scholars.

You can get most of the e-books here (post the book name or ISBN in the search box) and most of scientific papers here (post the DOI in the search box).

Other than that, you can look at the various Brilliant problems, notes, contests and sets (I suggest you also save some of them offline since Brilliant will delete all the notes and sets on this website in the near future). I'll link some here.

Brilliant Integration Contests

1a) Season 1, Part 1
1b) Season 1, Part 2
1c) Season 1, Part 3

2a) Season 2, Part 1
2b) Season 2, Part 2

3a) Season 3, Part 1
3b) Season 3, Part 2

Brilliant Summation Contests

1a) Season 1, Part 1
1b) Season 1, Part 2

Also, search problems and solutions by the following users on Brilliant (most of them are inactive, but they have posted plenty of stuff) : Mark Hennings, Hummus A, Aditya Narayan Sharma, Julian Poon, Ronak Agarwal, Anastasiya Romanova, Jake Lai, Kartik Sharma, Pi Han Goh, Aman Rajput, Aditya Kumar.

Finally, you can browse Math Stack Exchange website for more interesting problems in this realm. Some profiles which have interesting content for these topics are Robjohn, Sangchul Lee, Ron Gordon, M.N.C.E., Start wearing purple, Random Variable, Tunk-Fey, r9m, Achille Hui, Olivier Oloa, Marko Riedel, Mike Spivey, Markus Scheuer, Zaid Alyafeai, nospoon, Jack D'Aurizio, Felix Marin, Anastasiya Romanova, Cleo.

Thereafter, there are many more books/topics depending on your level, but this should suffice for now.

Ishan Singh - 2 years, 7 months ago

@Ishan Singh – @Ishan Singh Thanks a lot!!!!!! Like this information is AMAZING!!!! I am already about to complete Concrete Maths right now.......(got to know about it, only a few weeks earlier.!!) And, I have solved many questions from the integration contests, but I had never heard of Brilliant Summation Contests!!..........But yet again THANKS for all the references and especially the books.....!! I'm gonna start doing them from Nov 4th onwards......(Till then gotta prepare for KVPY....!!) But these are going to be real helpful....i can tell by the looks of them.......!!!!

Aaghaz Mahajan - 2 years, 7 months 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}`	$\boxed{123}$