Series: The Basics

A series is simply what you get when you add up all the numbers in a sequence, one after another, forever. The name can be confusing, since in everyday English "series" and "sequence" mean about the same thing, but in math a sequence is the list and a series is the sum of that list. Series are good for totals built from infinitely many small pieces, like the full distance a bouncing ball travels, and later in this course they'll let you build functions like $\sin x$ out of simple powers. The formula $\sum_{n=1}^{\infty} a_n = \lim_{n\to\infty} s_n$ looks heavy, but it's just a few pieces: $\sum$ (add things up), $a_n$ (the number you add at step $n$), $s_n$ (the running total after $n$ steps) and $\lim$ (where that running total is heading). It looks complex, but all it says is "the sum is wherever the running total ends up."

Nobody can literally do infinitely many additions. So we do what you'd do with a calculator: add a few, write down the total, add one more, write down the new total, and see where those totals are heading. The partial sums $s_1 = a_1$, $s_2 = a_1 + a_2$, $s_3 = a_1 + a_2 + a_3, \dots$ form a sequence, and that's the nice part: everything from the last two lessons applies to it. If that sequence converges to $s$, the series converges and its sum is $s$. Otherwise it diverges.

• $\sum \frac{1}{2^n}$: $s_n = 1 - \frac{1}{2^n} \to 1$. Converges to 1.
• $\sum n$: $s_n = \frac{n(n+1)}{2} \to \infty$. Diverges.
• $\sum (-1)^n$: $s_n$ alternates $-1, 0, -1, 0, \dots$ Diverges, even though it stays bounded.

A common mistake. There are two sequences here, and it's tempting to treat them as one. The terms $a_n$ are the pieces you add; the partial sums $s_n$ are the running totals. The series converges when the partial sums settle, not when the terms do. In $\sum \frac{1}{2^n}$ the terms head to 0 while the sum heads to 1. Two different limits, two different questions.

Index shifts. The index is a dummy variable, so $\sum_{n=3}^{\infty} \frac{1}{(n-2)^2} = \sum_{n=1}^{\infty} \frac{1}{n^2}$: replace $n$ by $n + 2$ and lower both limits by 2. You can also peel off terms: $\sum_{n=1}^{\infty} a_n = a_1 + a_2 + \sum_{n=3}^{\infty} a_n$.

Arithmetic. If $\sum a_n = A$ and $\sum b_n = B$, then $\sum c\,a_n = cA$ and $\sum (a_n \pm b_n) = A \pm B$. These come straight from the limit laws applied to the partial sums. Products are different: $\sum a_n b_n$ is not $AB$ in general, just like $(a_1 + a_2)(b_1 + b_2)$ isn't $a_1b_1 + a_2b_2$.

Repeating decimals, perpetuities in finance (a payment every year forever, each worth less today), and a digital filter's response to a single pulse are all infinite series, each defined as the limit of its partial sums.