Infinite series

What it is

A (real) infinite series is an expression of the form $$ a_1+a_2+a_3+\cdots, $$ where $a_i$'s are all real numbers. Of course, we cannot really add up infinitely many numbers (we shall die before it is over). So what we mean by this sum is $$ \lim_{n\rightarrow\infty} (a_1+\cdots + a_n). $$ The sum $a_1+\cdots+a_n$ is called the $n$-th partial sum.

If this limit exists, then we say that the series converges to that value. An example is $1+\frac 12+\frac{1}{2^2}+\frac{1}{2^3}+\cdots = 2.$
If the limit is $\infty $ or $-\infty $ then we say that the series diverges to $\infty $ or $-\infty.$ A trivial example is $1+1+1+\cdots = \infty.$
It is possible that the series neither converges nor diverges. We then say that the series oscillates. An example is $$ 1-1+1-1+1-1+-+-\cdots. $$ Here the partial sum are $$ 1,0,1,0,1,0,..., $$ which clearly oscillates.

The definition of an infinite series as the limit of its partial sums is a natural extension of the concept of usual addition. However, an infinite series may show certain counterintuitive behaviours, as we shall see next.

Strange things

Not associative

Ordinary addition is associative, i.e. you group the numbers as you please without affecting the sum. For instance, $(a+b)+(c+d) = (((a+b)+c)+d).$

Not so for an infinite series. For example, the oscillating series $$ 1-1+1-1+1-+-+\cdots, $$ when grouped like this $$ (1-1)+(1-1)+(1-1)+(-)+(-)+\cdots, $$ actually becomes $$ 0+0+0+\cdots, $$ which converges to $0.$ (Don't think of $0\times \infty=$ undefined. Here the partial sums are all $0,$ and so the limit of partial sums is $0.$) If you group the terms like $$ 1+(-1+1)+(-1+1)+()+()\cdots, $$ then you'll a get a series converging to $1.$

Not commutative

Changinf the order of the summands does not change a sum. This is called commutativity of addition. Unfortunately, this may fail for an infinite series.

Consider again our old friend $$ 1-1+1-1+1-+-+\cdots. $$ All the even terms are $1$ and all the odd terms are $-1,$ right? We shall rearrange these a bit to get $$ 1+1-1+1+1-1++-++-\cdots. $$ Don't think that we have increased the number of $1's.$ We still have countably infinitely many $1$'s and just as many $-1$'s.

This rearranged series has partial sums $$ 1,2,1,2,3,2,... $$ The pattern is simple it goes up 2 steps then comes down by 1. So effectively it is increasing, and increasing without bounds. In short, the rearranged series diverges to $\infty.$

Nice series

Absence of the helpful and familiar properties like associativity and commutativity makes it a bit difficult to work with infinite series. Thankfully, there are certain types of infinite series that behave nicely (i.e., regrouping and rearranging them do not change the sums). One type is mentioned in the next theorem.

Theorem An infinite series consisting of only nonnegative terms will never change the sum howsoever your regroup or rearrange the terms.

Thanks to this result, we never need to worry about adding infinitely many probabilities when using the 3rd axiom.

Later in this course, we shall need to work with infinite series with negative as well as positive terms. The following theorem gives us a condition under which these series will behave nicely.

Theorem Consider an infinite series $a_1+a_2+a_3+\cdots,$ where $a_i$'s are any real numbers (positive/negative/zero). If the infinite series of the absolute terms $|a_1|+|a_2|+|a_3|+\cdots$ converges (to some finite limit), then the terms in the original series may be regrouped and rearranged without affecting the sum.

If this condition is met (i.e, the absolute series converges), then we call the original series absolutely convergent. It is an interesting and very useful fact that any absolutely convergent series must also be convergent.

You'll learn the proofs of all these theorems in you analysis course.