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 xN, 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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5 Responses to “Monday Math 25”

  1. Monday Math 26 « Twisted One 151’s Weblog Says:

    […] is , thus telling us . Note that this sum is , where is the Riemann zeta function introduced in this previous post. The problem of finding the exact value of this series and proving it was known as the Basel […]

  2. Monday Math 33 « Twisted One 151’s Weblog Says:

    […] for -1<x≤1. Setting x=1 tells us that the harmonic series converges to . Now, recall the Riemann Zeta Function, which for is given by . Suppose we define an analogous function with alternating terms: . This […]

  3. Monday Math 58 « Twisted One 151’s Weblog Says:

    […] via the geometric series, . Thus Note that we can reverse the order of summation: , where is the Riemann zeta function. Now, we use Euler’s formula: as , , , and . Previously, we showed that the function can be […]

  4. Monday Math 59 « Twisted One 151’s Weblog Says:

    […] Math 59 By twistedone151 Recall the Riemann zeta function . Now, consider the product . We see: , and we have the zeta function […]

  5. Monday Math 87 « Twisted One 151’s Weblog Says:

    […] if we take the trivial sequence 1,1,1,… (an=1 for all n), then the Dirichlet series is the Riemann zeta function . For the alternating sequence 1,-1,1,-1,… (an=(-1)n-1), the Dirichlet series is the […]

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