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Harmonic Series in Math: Definition & Formula

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  • 0:03 Counterexamples and…
  • 1:27 Proof of Divergence
  • 3:49 Lesson Summary
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Lesson Transcript
Instructor: Ryan Hultzman
The harmonic series provides one of the most important counter-intuitive examples in the study of mathematics. In the harmonic series, the numbers or terms get closer and closer to zero, while the series itself diverges.

Counterexamples and the Harmonic Series

A mathematical series is the sum of all the numbers, or terms, in a mathematical sequence. A series converges if its sequence of partial sums approaches a finite number as the variable gets larger or smaller. Conversely, a series diverges if its sequence of partial sums does not approach a finite number. The limit of its partial sums in a divergent series may be nonexistent; it may also be positive or negative infinity.

One of the easiest and most useful tests you can use to determine if a series diverges deals with the limit of the terms in the series. You may have learned that if the limit of the terms of the series is not zero, then the series must diverge. But what if the limit of the terms of the series is indeed zero? Does this imply that the series must always converge? There are many examples of this being true - namely the geometric series. Are there any examples of when this is false?

Enter the harmonic series. The harmonic series is the sum from n = 1 to infinity with terms 1/n. If you write out the first few terms, the series unfolds as follows: 1 + 1/2 + 1/3 + 1/4 + 1/5 +. . .etc.

As n tends to infinity, 1/n tends to 0. However, the series actually diverges. Even though we are going to be adding extremely tiny numbers of 1/n, as n gets gigantic, our partial sums will still be growing without bound.

Proof of Divergence

Let's take a look at a proof that can show us that the harmonic series is indeed divergent. We'll start by playing devil's advocate and supposing that the harmonic series is convergent. By making this assumption, our goal is to create some sort of clear contradiction. We'll use this contradiction to show that our initial assumption, the convergence of the harmonic series, must be false, and the harmonic series is divergent. Since we're assuming that the harmonic series is convergent, let's suppose that the value of the series is equal to some finite number H.

That is, H = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + . . .etc.

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