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College Algebra: Help and Review27 chapters | 230 lessons | 1 flashcard set

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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.

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.

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.

- Let's manipulate the right hand side of the sum by making it smaller. We know that 1/3 > 1/4, so let's replace the 1/3 in our sum with a 1/4, while taking into account the fact that our sum got smaller. Hence, H > 1 + 1/2 + 1/4 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + . . .etc.
- Similarly, 1/5 > 1/6, 1/7 > 1/8 and so forth. By replacing these larger numbers with the smaller numbers, we see that the right hand side keeps getting smaller, so H > 1 + 1/2 + 1/4 + 1/4 + 1/6 + 1/6 + 1/8 + 1/8 + . . .etc.
- Now, let's add 1/4 + 1/4, 1/6 + 1/6, 1/8 + 1/8 and so forth. We then see that H > 1 + 1/2 + 2/4 + 2/6 + 2/8 + . . .etc.
- Reducing these fractions to their lowest form, we get H > 1 + 1/2 + 1/2 + 1/3 + 1/4 + . . .etc.
- We can switch the place of the 1 with the first 1/2 to get H > 1/2 + 1 + 1/2 + 1/3 + 1/4 + . . .etc.

Notice that in this sum, we have 1 + 1/2 + 1/3 + 1/4 + . . ., which is the harmonic series that we assumed has a sum equal to H. We can substitute an H for the sum and see that H > 1/2 + H. Subtracting H from both sides, we see that 0 >1/2 - but that's absurd. This is the contradiction we were hoping to find. It implies that our original assumption, that the harmonic series is convergent, must be false. Hence, the harmonic series is divergent.

The harmonic series is important because it provides a simple counterexample to the claim, 'if the limit of the terms in the series is zero, then the series must converge.' Remember, the harmonic series diverges even though the limit of the terms in the series is zero. In a **harmonic series**, the numbers, or terms, get smaller, while the sum of the series gets larger. Understanding the properties of the harmonic series is key to understanding the core concepts behind series and infinite sums.

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College Algebra: Help and Review27 chapters | 230 lessons | 1 flashcard set

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