The following equation is almost true.

And by almost true, I mean correct to well over 200 decimal places. This sum comes from [1]. Here I will show why the two sides are very nearly equal and why they’re not exactly equal.

Let’s explore the numerator of the sum with a little code.

>>> from math import tanh, pi >>> for n in range(1, 11): print(n*tanh(pi)) 0.99627207622075 1.9925441524415 2.98881622866225 3.985088304883 .... 10.95899283842825

When we take the floor (the integer part [2]) of the numbers above, the pattern seems to be

⌊*n* tanh π⌋ = *n *− 1

If the pattern continues, our sum would be 1/81. To see this, multiply the series by 100, evaluate the equation below at *x* = 1/10, and divide by 100.

Our sum is close to 1/81, but not exactly equal to it, because

⌊*n* tanh π⌋ = *n − *1

holds for a lot of *n*‘s but not for all *n*.

Note that

tanh π = 0.996… = 1 − 0.00372…

and so

⌊*n* tanh π⌋ = *n *− 1

will hold as long as *n* < 1/0.00372… = 268.2…

Now

⌊268 tanh π⌋ = 268 − 1

but

⌊269 tanh π⌋ = 269 −2.

So the 269th term on the left side

is less than the 269th term of the sum

10^{−2} + 2×10^{−3} + 3×10^{−4} + … = 1/81

for the right side.

We can compare the decimal expansions of both sides by using the Mathematica command

N[Sum[Floor[n Tanh[Pi]]/10^n, {n, 1, 300}], 300]

This shows the following:

## Related posts

[1] J. M. Borwein and P. B. Borwein. Strange Series and High Precision Fraud. The American Mathematical Monthly, Vol. 99, No. 7, pp. 622-640

[2] The floor of a real number *x* is the greatest integer ≤ *x*. For positive *x*, this is the integer part of *x*, but not for negative *x*.

You may find redpenblack’s video proving the sum converges to 1/81 worth sharing.

https://youtu.be/opeW_1aG2sU