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Summation Proof that the Harmonic Series Diverges Begin with a square with each side equal to one-unit in length (Area = 1 unit2) and create a second square of equal area by reflection. Add half the new square's area to the initial area, thus the new total area is 1½ units2. Add to that total area half-of-the-half square (one-quarter) to get 1¾ units2 of area. Add to that half of the quarter square (one-eighth) to get 1.875 units2of area. If this process is continued for infinitely many steps, the total area approaches the full area of the two squares for a total area of 2 units2. Visually, the first five steps of this process is shown below.
This addition of areas can be expressed with "summation notation" as follows:
This summation is called an infinite series. Other examples of infinite series include:
[P is a prime number]
In each of these examples the sum is infinity. This means that if you continue the sequence of terms forever and add all the terms, the sum is infinitely large. It grows without bounds. However, there are series that do not grow beyond all bounds. Many series have a boundary that marks the limit of their growth. The series of powers of ½ is an example of this type of series with a limit and is called the "Harmonic Series."
There is a formula for the sum of an infinite series:
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In the case of the series
, the first term is 1 and the ratio of any pair of consecutive terms is ½. The ratio is found by taking any term and dividing it by the term that came immediately in front of it. Thus,
,
,
, and so on.
Another way to look at the "common ratio" is to say that each new term is found by multiplying the last term by the same number. Thus,
,
,
,
, and so on.
Using the above formula for the sum of the infinite series, we see that the sum is indeed 2:
Before you assume that any series of fractions smaller than 1 will have a limit, look at the following series that increases without bounds:
Why should the sum of the powers of ½ have a limit but the sum of the fractions "one-over-n" increase without bounds? The proof is rather clever.
Proof :
The first term is equal to one. Now note the next three terms:. The sum of these three terms is definitely greater than one:
. Next, observe that the next four terms have a sum greater than ½ and then the next eight terms also will sum greater than ½, thus together the terms 5 through sixteen have a sum greater than 1.
This pattern will continue forever. We can continue to group terms that will sum greater than one and if we add an infinite string of ones the ultimate sum will increase without bounds. Why does this work? Note that the terms get smaller as we move to the right. Thus, if we look at any term in the series, all the terms that came before it will be greater than that term. Thus, the three terms before ¼ are all greater than ¼. It is obvious that four terms of each ¼ will add to one,
,
thus, if we add four numbers that are greater than or equal to ¼ the sum will be greater than or equal to 1. The four terms in position five through eight are all greater than or equal to one-eighth and thus they will sum to something greater than ½. The eight terms from position 9 through 16 are all greater than or equal to one-sixteenth and thus will sum to something greater than ½ [eight times one-sixteenth]. Thus, terms 5 through 16 sum to something equal to or greater than one. Since the series is infinite we can repeatedly go out far enough to get more ones. Thus, the sum increases without bounds.
This topic, the limits of infinite series, is one of the most fascinating topics in all of mathematics. The study of it falls under the field called "Analysis" (or "Real Analysis").
.
In this case the sum is 
, where
a1 is the first term of the series and r
is the common ratio between any pair of terms.
