Consider this thought experiment. Suppose that you ask the first person you meet for one dollar. Ask the second person you meet for half of a dollar. Ask the third person you meet for one-third of a dollar. To the one-hundredth person you meet, you ask for one-hundredth of a dollar. To the one thousand and first person you meet, you ask for $\frac{1}{1001}$ of a dollar and so on. Suppose that it is possible to break down a dollar to a fraction as tiny as possible (and that the person you ask will give the money to you). Suppose that there are unlimited number of people for you to ask. If you keep asking and receiving in this manner, how much money will you have?

Mind you that the amounts you ask and receive are getting smaller and smaller to the point that it is practically zero. From the 100th person, you get one cent. The amount you get from a person beyond the first one hundred seems to be trivially small. From the first 100 people, the total amount would be about $5.19. Is it possible that going beyond 100 people might not net you much more that five dollars? You may want to think about the puzzle. Here’s one way to look at the thought experiment. From the first person, you get$1. From the second person, you get $\frac{1}{2}$ of a dollar. From the next two people, you get $\frac{1}{3}+\frac{1}{4}$, which is greater than $\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$. Now from the next 4 people, you get $\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}$, which is greater than $\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{1}{2}$. From the next 8 people, you get

$\displaystyle \frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{1}{15}+\frac{1}{16}$,

which is greater than

$\displaystyle \frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}=\frac{1}{2}$.

And so on. Each group of people you ask is twice as big as the previous group. The amount of money from each group is always $\frac{1}{2}$ of a dollar or more. Indeed, if you can ask enough groups as described above, you can have as much money as you would like. Remember, the first group has one person (giving you \$1), and the second group has one person (giving you half a dollar). The third group has 2 people and the fourth group has 4 people. Each subsequent group has twice as many people as the previous group, but each group will net you at least half a dollar. To get one million dollar, you ask two million groups. With each group giving you at least half a dollar, you will get at least one million dollars in total. Of course, the total number of people in these 2 million groups would be astronomical! To get one trillion dollars, you would ask 2 trillion groups of people. The idea here is that if you want $N$ dollars, you can receive more than that amount by asking enough groups of people (i.e. by asking enough people). In fact this is like a definition of infinity. You know, for a certain target, no matter how big it is, you can always reach that target by performing enough iterations. Then mathematically speaking, it is a limitless phenomenon. So we can write the following:

$\displaystyle 1+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \cdots + \frac{1}{n}+ \cdots = \infty$

$\displaystyle \sum \limits_{n=1}^\infty \frac{1}{n}=\infty$

The above are two different ways to express an infinite series. The first way is to write out the sum of the first several terms and then to follow that by dot dot dot. The second way uses the uppser case Greek letter Sigma, which denotes sum. So it is saying that it is a sum of $\frac{1}{n}$ as $n$ starts from 1 and increases without bound (or from 1 to infinity). The terms in the sum are getting smaller and smaller to become so small it is practically and essentially zero (a more economical way of saying that is that the limit of the term $\frac{1}{n}$ is zero). Here, we are summing numbers that are close to zero and yet the sum is infinity. Another way to say it is that this infinite series diverges. This may seem counter intuitive, but the math is pretty clear if you think about it and let it sink in.

The above infinite series goes by the name of harmonic series. The name has a musical connection; it derives from the concept of overtones or harmonics in music. Here, we stick with the mathematical side of things. So the harmonic series diverges (or not converges to a finite number).

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Another Proof

The above proof that the harmonic series is a classical proof. It was first given in the 14th century. Anyone who wants to study the concept of infinite series should learn this proof. However, it requires the mental leap of grouping the terms in groups of exponential sizes (2, 4, 8, 16, and so on). Here’s another proof that requires only grouping two terms at a time.

$\displaystyle S=1+\frac{1}{2}+\biggl(\frac{1}{3}+\frac{1}{4}\biggr)+\biggl(\frac{1}{5}+\frac{1}{6}\biggr)+\biggl(\frac{1}{7}+\frac{1}{8}\biggr)+\cdots$

which is greater than

$\displaystyle 1+\frac{1}{2}+\biggl(\frac{1}{4}+\frac{1}{4}\biggr)+\biggl(\frac{1}{6}+\frac{1}{6}\biggr)+\biggl(\frac{1}{8}+\frac{1}{8}\biggr)+\cdots$

$\displaystyle =1+\frac{1}{2}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots=\frac{1}{2}+S$

If $S$ were finite, then $S>\frac{1}{2}+S$, which is a contradiction. So $S$ must be infinite. This proof is an old one too. It is a clever proof and is just as valid mathematically as the first one. Both proofs are useful in helping us understand the harmonic series.

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Other Sequences

To contrast, what if we sum not all the terms $\frac{1}{n}$? What is we just sum a subset? Here’s are several examples.

$\displaystyle 1+ \frac{1}{2^1}+ \frac{1}{2^2}+ \frac{1}{2^3}+ \cdots + \frac{1}{2^n}+ \cdots$

$\displaystyle 1+ \frac{1}{2^2}+ \frac{1}{3^2}+ \frac{1}{4^2}+ \cdots + \frac{1}{n^2}+ \cdots$

$\displaystyle \frac{1}{2}+ \frac{1}{3}+ \frac{1}{5}+ \frac{1}{7}+ \frac{1}{11}+ \frac{1}{13}+\frac{1}{17}+ \cdots$

Which of the above series converge and which diverge? In the first series, the denominator in each term is a power of 2. In the second series, the denominator in each term is the square of an integer. In the third, the denominator of the terms ranges over all the prime numbers.

The harmonic series is a great example to introduce the concepts of series and infinite series. But it is just a beginning. We do not intend to delve too deeply into the subject. For more information about harmonic series, see the Wikipedia entry. Here’s the the Wikipedia entry on series. This is a concept that is both wide and deep in mathematics. Google online or find some good math books. A good book on infinite series and related topics is Principles of Mathematical Analysis by Walter Rudin.

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$\copyright \ 2016 \text{ by Dan Ma}$