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Numbers Defined By Infinite Sums

Asked by a student at Mission Bay High School on September 21, 1997:
What if someone did this : k = 1/sqrt(1) + 1/sqrt(2) ... and it comes to some amount. Would that number be "special"? Are there any other special constants that haven't been discovered?
This particular sum does not add up to any (finite) number. (We say that the sum diverges; you can also think of it as "adding up to infinity").

To see this first note that 1/sqrt(1) + 1/sqrt(2) + 1/sqrt(3) + ... is greater than 1/1 + 1/2 + 1/3 + .... The second sum is often called the harmonic series and is known to diverge. This is because we can regroup the terms and write it as (1/1) + (1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ... . Now this sum is greater than (1/1) + (1/2) + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + ... . Note that each parenthesized group except the first adds up to 1/2. So, after the first group, the sum is at 1. After the next two groups, the sum as at 1 + (1/2) + (1/2) = 2. After the next two groups, the sum is at 2 + (1/2) + (1/2) = 3. And so on. Continuing in this way, we see that the sum eventually passes every finite number, so the total sum is infinite.

In general there are many numbers which are defined by infinite sums. For instance e = 1 + 1/2! + 1/3! + 1/4! + ... where n! = (n)(n-1)(n-2)...(2)(1). Pi can also be expressed as a sum, although it is a bit more complicated.

Your example can be modified to add up to a finite number. For each power p > 1, the sum 1/1^p + 1/2^p + 1/3^p + ... does add up to a finite value. But no special name is given to that value.

There are infinitely many real numbers (in fact, what mathematicians call "uncountably infinitely many" which is even more than just "infinitely many"!), so there are plenty of numbers that nobody has had occasion to think about in a special way. That doesn't mean they haven't been "discovered", though; after all, until now nobody has probably ever before written down the number

but that doesn't mean I somehow "discovered" it by writing it down for the first time. A "discovery" would be discovering that some number had special, unique, and important properties that no other number has.

It is difficult to say which numbers are "special." Numbers such as pi and e are special because they occur naturally in many situations in applied mathematics. There are a variety of other constants in mathematics which are somewhat less useful but nevertheless still relevant to some people.

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