Close but no cigar
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 = 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 = n - 1
holds for a lot of ns but not for all n.
Note that
tanh = 0.996... = 1 - 0.00372...
and so
n tanh = n - 1
will hold as long as n < 1/0.00372... = 268.2...
Now
268 tanh = 268-1
but
269 tanh = 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:
[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.
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