It is frequently noted that the harmonic series diverges. One way to demonstrate this is the integral test:

Given a non-negative, monotonically decreasing function defined for *x*≥*N*, *N* some integer, the series converges if and only if the integral converges (is finite); if the integral diverges, so does the sum.

For the harmonic series, is such a function for *x*≥1, so we check the integral . As , and , the integral diverges, and thus the harmonic series diverges.

However, one might recall from calculus that converges for *p*>1. Thus, the series converges for *p*>1. In fact, this is a continuous function of *p*, and the series also converges when the power is a complex number *s* with . The analytic continuation of this function to the entire complex plane, save the simple pole at *s*=1, is called the Riemann zeta function, and is written . The Riemann zeta function is central to one of the most important unsolved problems in mathematics, the Riemann hypothesis (it is one of the seven Millenium Prize Problems).

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Tags: Convergence, Harmonic Series, Integral Test, Math, Monday Math, Riemann Zeta Function, Series

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