Recently I was trying to find infinite series expressions for some famous mathematical constants, and I thought I would record what I found here. This post considers , Euler’s constant e, the golden ratio , and the Euler-Mascheroni constant .
(Of course, you can construct an infinite series for any of these constants by simply adding the next significant digit in the constant’s infinite decimal representation with each successive term in the series, but that’s kind of boring.)
First, . An infinite series representation for starts with the geometric series formula , valid for . If you replace by , you get
Integrating both sides of this expression from 0 to x yields
as long as the infinite series on the right converges (which it will when , thanks to the alternating series test).
Substituting 1 for x yields
Then, multiplying both sides by 4 produces the Leibniz series for :
Second, . We’ll take as our definition of e that e is the value of a such that . Constructing the Maclaurin series for and substituting 1 for x will give us an infinite series expression for e.
Since our definition of is such that regardless of the value of n, the Maclaurin series for is simply
Substituting 1 for x gives us
(The fact that the Maclaurin series for converges for any real value of x is an easy application of the ratio test, and the fact that the Maclaurin series for actually converges to follows from a slightly more involved application of the Lagrange Remainder Theorem.)
Third, . The first two infinite series are well-known, but the one we give here for the golden ratio is (I believe) much less so. It starts with a property of that is fairly well-known, namely, that the ratio of the Fibonacci number to the th Fibonacci number approaches as n approaches infinity:
To obtain an infinite series expression for , then, we just need to represent as the partial sum of an infinite series. Well, this telescoping sum works, although it’s rather boring:
However, grouping terms differently gives us something that is perhaps interesting. For example, we have
, where the last step is due to Cassini’s identity.
Grouping terms this way leaves out , so we have
(I found this one from Thomas Andrews’s answer here on Math.SE.)
Finally, . Our infinite series expression for is derived in a manner similar to our series for : We’ll take a well-known expression for and turn it into an infinite series.
The usual definition of is that it is the long-term error in the approximation , where is the th harmonic number . Our goal, then, is to turn into the partial sum of an infinite series. Here’s one such, although (as in our example with ), it’s rather boring:
Thanks to properties of logarithms, though, this can be expressed more interestingly as
By interspersing the terms in this expression with the terms in and taking the limit, we have the following infinite series expression for :