Sum of Harmonic Series
The above series is known as Harmonic Series.
It can be shown to diverge using the integral test by comparison with the function 1/x
The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries. The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter.
The sum is given as below
It can be shown to diverge using the integral test by comparison with the function 1/x
The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries. The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter.
The sum is given as below
Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2, and since an infinite sum of 1/2's diverges, so does the harmonic series.
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