Series

Frequently in analysis it is useful to consider the sum of infinitely many numbers. So for some sequence of numbers (a_n) we may want to consider expressions like

$\sum _{n=1}^{\infty }a_{n}$.

But the exact meaning of this may not be immediately clear. Intuitively we would like to say that the sum of infinitely many numbers should be the number we get close to after we have summed a large number of terms. We will use the notion of the limit of a sequence to make this precise. The standard terminology in the study of series sometimes has room for improvement, but we follow the standard terminology in this section.

Definition

We begin with a sequence (a_n) of the numbers we would like to sum.

Definition A series of real numbers is an infinite formal sum

$\sum _{n=1}^{\infty }a_{n}$

where each term a_n is a real number.

This definition deserves a few comments. The first is that no attempt is being made to define a formal sum. It is possible to define a formal sum simply as the sequence of terms, but this doesn't add any clarity to the discussion. The reason for allowing the series to be formal is merely a matter of convenience. It is frequently easier to refer to a series before it as been determined if there is any number that should represent the sum. This is very similar to the standard practice of saying $\textstyle \lim _{n\to \infty }a_{n}$ does not exist - we have only defined the meaning of the symbols $\textstyle \lim _{n\to \infty }a_{n}$ in the case when the limit does exist. We should instead say that the sequence a_n does not converge. However, the meaning of the previous statement is perfectly clear.

Definition The n-th partial sum of a sequence a_n is defined to be the sum of the first n terms of (a_n), that is

$S_{n}=\sum _{i=1}^{n}a_{i}=a_{1}+a_{2}+\cdots +a_{n}$.

When the sequence a_n is being thought of as the terms of a series, then S_n is often called the n-th partial sum of the series.

Often several partial sums may appear in the same argument, so the partial sum is often written simply as $\sum _{i=1}^{n}a_{i}$ instead of S_n when we wish to avoid confusion.

Definition For a series $\textstyle \sum _{n=1}^{\infty }a_{n}$ we define the sum of the series to be the limit of the partial sums. That is, we define:

$\sum _{n=1}^{\infty }a_{n}=\lim _{N\to \infty }\sum _{n=1}^{N}a_{n}=\lim _{N\to \infty }S_{N}$.

If the limit exists we say the series converges, otherwise we say the series diverges.

It should be emphasized when a series diverges, we cannot interpret $\textstyle \sum _{n=1}^{\infty }a_{n}$ as a number, but only as a formal sum. On the other hand when the series does converge, it is often not known what number the series converges to, so this number is usually denoted by $\textstyle \sum _{n=1}^{\infty }a_{n}$. Again this is similar to writing $\textstyle \lim _{n\to \infty }a_{n}$ before the convergence of the sequence a_n is established. In practice the meaning of the symbols $\textstyle \sum _{n=1}^{\infty }a_{n}$ is clear from context.

Often it is convenient to sum series starting from some number other than n = 1, and start the series some other point like 0, 2, −10, etc. This hopefully should cause no confusion; the sum of the series is still defined as the limit of the partial sums. Often, it is clear from context where the sum begins. In these cases it is not uncommon to leave the index out of the sigma notation - that is, it is useful to write ∑a_n for $\textstyle \sum _{n=1}^{\infty }a_{n}$

Examples

• The notion of an infinite sum can be a little more subtle than it first appears. For example, consider the sequence a_n = (−1)ⁿ. Does the sum of the a_n converge or diverge? We could consider the partial sums S_N, which are 1 if N is odd and 0 if N is even. So it seems the series diverges. On the other hand, 1 − 1 + 1 − 1 + 1 − 1 + … = (1 − 1) + (1 − 1) + (1 − 1) + … = 0 + 0 + 0 + … = 0. Does then the series diverge or converge to 0, according to our theory? The answer is it diverges; the fallacy in the previous sentence was in asserting that 1 − 1 + 1 − 1 + 1 − 1 + … = (1 − 1) + (1 − 1) + (1 − 1) + …. It is no longer true that for infinite series that we may sum in any order we choose; we must justify why we are allowed to group the +1's together with the -1's without changing the sum. Stated formally, associativity does not necessarily hold for infinite series. We will investigate when it is possible to rearrange the elements of a series and still get the same sum.
• Perhaps the most familiar examples of series come from decimal expansions of real numbers. While we have not given a rigorous definition of decimal expansions here, it can be shown that every real number r can be expressed in the form $\textstyle r=\sum _{n=0}^{\infty }{\frac {a_{n}}{10^{n}}}$ where a_n ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
• Consider the series $\textstyle {\tfrac {1}{1\cdot 2}}+{\tfrac {1}{2\cdot 3}}+{\tfrac {1}{3\cdot 4}}\cdots =\sum _{n=1}^{\infty }{\tfrac {1}{n(n+1)}}$. At first it may seem difficult to determine the partial sums and decide whether or not the series converges. However, it turns out that this series is nicer than it first appears, since $\textstyle {\frac {1}{n(n+1)}}={\frac {1}{n}}-{\frac {1}{n+1}}.$ Thus
  
  

  $\sum _{n=1}^{N}{\frac {1}{n(n+1)}}=\sum _{n=1}^{N}{\frac {1}{n}}-{\frac {1}{n+1}}=1-{\frac {1}{2}}+{\frac {1}{2}}-{\frac {1}{3}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots +{\frac {1}{N}}-{\frac {1}{N+1}}=1-{\frac {1}{N+1}}.$

  
  Thus $\textstyle \lim _{N\to \infty }\sum _{n=1}^{N}{\tfrac {1}{n(n+1)}}=\lim _{N\to \infty }(1-{\tfrac {1}{N+1}})=1.$ This is an example of a telescoping series - that is, a series that can be written in the form such that the next term cancels with the previous terms. That is,

  $\sum _{n=1}^{\infty }a_{n}-a_{n+1}$

  For all such series the partial sums are given by S_N = a₁ − a_N+1. That is, the terms of the series collapse like a telescope, leaving only the first and last. Such a series converges to a₁ − lim a_N+1.
• Another important example of a series is the geometric series given by $\sum _{n=0}^{\infty }r^{n}$. The partial sums for the geometric series first appear quite complicated. At first glance the partial sums would just appear to be S_N = 1 + r +r² + … + r^N, but if we calculate (1 − r)S_N then the sum telescopes and we are left with (1 − r)S_N = (1 − r)^N+1. Notice that this sum holds for any r or N, so we conclude that $\textstyle S_{N}={\tfrac {1-r^{N+1}}{1-r}}$. Thus a geometric series converges if |r| < 1.