Close but no cigar

The following equation is almost true.

\sum_{n=1}^\infty \frac{\lfloor n \tanh \pi\rfloor}{10^n} = \frac{1}{81}

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.

\begin{align*} \sum_{n=1}^\infty n x^{n-1} &= \sum_{n=0}^\infty \frac{d}{dx} x^n \\ &= \frac{d}{dx} \sum_{n=0}^\infty x^n \\ &= \frac{d}{dx} \frac{1}{1-x} \\ &= \frac{1}{(1-x)^2} \end{align*}

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

\sum_{n=1}^\infty \frac{\lfloor n \tanh \pi\rfloor}{10^n} = \frac{1}{81}

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:

\begin{align*} \frac{1}{81} &= 0.\underbrace{012345679}_{\text{repeat forever}} \ldots\\ \sum_{n=1}^\infty \frac{\lfloor n \tanh \pi\rfloor}{10^n} &= 0.\underbrace{012345679}_{\text{repeat 29 times}} \ldots 0123456\framebox{6}\ldots \end{align*}

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.

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