Douglas Hofstadter discovered that the 8th harmonic number equals *e*.

OK, not really. The following equation cannot possibly be true because the left side is rational and the right side is irrational.

However, Hofstadter showed that the equation **does** hold if you carry all calculations out to three decimal places.

1.000
0.500
0.333
0.250
0.200
0.167
0.143
0.125
-----
2.718

The following Python code gets the same result using four-place decimal arithmetic.

Here’s Python code to verify it.

>>> from decimal import *
>>> getcontext().prec = 4
>>> sum(Decimal(1)/Decimal(k) for k in range(1, 9))
Decimal('2.718')

## Related posts

The sum of the LHS fractions is 2 + 201/280.

If you look at the first eight terms of the continued fraction expansion of

e = [2; 1, 2, 1, 1, 4, 1, 1]

you get 2 + 51/71 = 2 + 204/284.

Because 201/280 and 204/284 are very close, the LHS sum is close to the eight-term continued fraction approximation.

Of course, the continued fraction is the better approximation to e.