Question Corner and Discussion Area

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

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

2374348627846827468726482678346278348236but 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.

This part of the site maintained by (No Current Maintainers)

Last updated: April 19, 1999

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

Navigation Panel:

Go backward to Finding the Ratio from the Sum of a Geometric Series

Go up to Question Corner Index

Go forward to A Sequence Describing a Bouncing Ball

Switch to graphical version (better pictures & formulas)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network Home Page