Evaluating Euler’s Number and Pi π with Series

Series Containing Factorials
Series that are neither arithmetic nor geometric must be considered individually. Often the limit of a series can be found to be finite by comparing the terms of the series to the terms of another series with a known limit. One such series

sum of factorials

Recall factorials from previous courses. We write n factorial as follows:



Consider the series

sum of factorials a

This series can be shown to have a limit by comparing its terms to the terms of the geometric series with a1=1 and r=½. Each term, an (n>1), of this geometric series is 1/2(n-1).

Previously, we found that for the geometric series with a1=1 and r=½, the series approaches 1/(1-½) or 2 as n approaches infinity. Therefore, ∑n=1 1/n!<2. Thus, this series is bounded above by 2.

To find a lower bound, notice that the series 1/1!+1/2!+1/3!+⋯ is the sum of positive terms. Therefore, for n>3,

Putting it all together, what we have shown is that for the series ∑n=1 1/n!

The Number e
Let us add 1 to both sides of the inequality derived in the previous section. We find that as n approaches infinity,

(A link to prove that 0!=1.)

The infinite series

is equal to an irrational number that is greater than 2½ and less than 3. We call this number e.

Eighteenth-century mathematicians computed this number to many decimal places and assigned to it the symbol e just as earlier mathematicians computed to many decimal places the ratio of the length of the circumference of a circle to the length of the diameter and assigned the symbol π to this ratio. Therefore, we say:

This number has an important role in many different branches of mathematics. A calculator will give the value of e to nine decimal places. As the result,
euler b

Evaluating e and π with series
Some infinite series can help us to evaluate important mathematical constants. For example, consider the series

euler c

Written out term by term, this series is
euler d

If we use a calculator to work out the first few partial sums of this series, we get
(1, 2, 2.5, 2.66667, 2.70833, 2.71667, 2.71806, …),

where we have written down some of the terms to five decimal places.

Now you might have noticed that this sequence of partial sums seems to be getting closer and closer to the number e, which is 2.71828 to five decimal places. In fact it can be shown that the partial sums do tend to e. So working out the partial sums of this series is a useful way of calculating e to a large number of decimal places.
Now let us look at the infinite series

evaluating pi a

For this series, we need to recall the meaning of the power (-1)(k+1). If k is odd then k+1 is even, and so (-1)(k+1)=1. On the other hand, if k is even then k+1 is odd, and so (-1)(k+1)=-1. We can now write out the series term by term as
evaluating pi b

Again we can use a calculator to work out the first few partial sums of this series. We get
(4, 2.6667, 3.4667, 2.8952, 3.3397, 2.9760, 3.2837, …)

where we have written down some of the terms to four decimal places.

This sequence of partial sums looks like it might be getting close to some number just greater than 3. In fact it can be shown that the partial sums tend to π, which is 3.1416 to four decimal places. If we kept on calculating the partial sums for this series, we would eventually obtain a value for π to several decimal places.

Key Point: Some infinite series can help us evaluate numbers like π and e as accurately as we choose.