A series is what you get when you add up the terms of a sequence. If your sequence is \(a_1, a_2, a_3, \ldots\), the corresponding series is \(a_1 + a_2 + a_3 + \cdots\), written compactly as \(\sum_{n=1}^{\infty} a_n\).
A finite series adds a fixed number of terms and always produces a definite number. An infinite series adds infinitely many terms — and the result might be finite (convergent) or might blow up to infinity (divergent). This surprising fact — that you can sometimes add infinitely many positive numbers and get a finite answer — is one of the most beautiful ideas in mathematics.
To walk 1 metre, you first walk ½ metre, then ¼, then ⅛, then 1/16... Adding infinitely many steps: \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1\). You do reach the destination. An infinite number of terms can sum to a finite value — and this is precisely the geometric series formula at work.
Define \(S_n = a_1 + a_2 + \cdots + a_n\) (the sum of the first \(n\) terms). The series converges if and only if the sequence of partial sums \(S_1, S_2, S_3, \ldots\) converges to a finite limit. Watch how \(S_n\) behaves as \(n\) grows — that's everything.
| Sigma notation | \(\displaystyle\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots\) |
| Geometric series sum | \(\displaystyle\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\quad(|r|<1)\) |
| Arithmetic series sum | \(S_n = \dfrac{n}{2}(a_1 + a_n)\) |
| Harmonic series | \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}\) — diverges (despite terms → 0) |
| p-series | \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^p}\) — converges iff \(p > 1\) |
| Partial sum notation | \(S_n = \displaystyle\sum_{k=1}^{n} a_k\) |
\(= \dfrac{1}{1-\frac{1}{2}} = 2\)
Divergence test → diverges
Famous result: \(= \dfrac{\pi^2}{6}\)
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