Sequences & Series

∑ notation and the closed forms: _i=1^n i = n(n+1)2 and geometric _k r^k = r^n-1r-1.

The idea

A sequence is an ordered list of numbers $a_{1}, a_{2}, a_{3}, \dots$, one for each position — a payment for each month, a population for each hour. A series is the sum of a sequence's terms: $2, 5, 8$ is a list, while $2 + 5 + 8$ is a single number. Summation notation abbreviates the sum: $\sum_{i=1}^{n} a_{i} = a_{1} + a_{2} + \cdots + a_{n},$ which we read as the sum of $a_{i}$ as $i$ runs from $1$ to $n$; the index $i$ takes each whole value from the lower limit to the upper one.

Two kinds of sequence occur constantly. A sequence is arithmetic when each term is the previous term plus a fixed common difference $d$, as in $2, 5, 8, 11, \dots$ with $d = 3$; it grows by repeated addition, like a linear function. It is geometric when each term is the previous term times a fixed common ratio $r$, as in $3, 6, 12, 24, \dots$ with $r = 2$; it grows by repeated multiplication, like an exponential function. Each kind has a closed form for its sum: a formula that gives the total directly, however many terms there are.

Ways to work on it

• Walkthrough. Summation notation, the Gauss sum, and the geometric-series formula.
• Practice. Evaluate arithmetic, square, and geometric sums via their closed forms.
• Hardest. Infinite geometric series, and the exact value of a repeating decimal.

Not sure where to start? Take the ten-question placement test.