Monday Math 25
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 \,dx)
converges (is finite); if the integral diverges, so does the sum.
For the harmonic series, =\frac{1}{x})
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 \gt1)
. 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).
Tags: Math, Monday Math, Series, Convergence, Integral Test, Harmonic Series, Riemann Zeta Function
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June 30, 2008 at 12:12 am
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August 18, 2008 at 12:07 am
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