Series Notation and Indexing

A series is a sum of the terms of a sequence. Sigma notation writes it compactly:

$$\sum_{k=1}^{n} a_k = a_1 + a_2 + \dots + a_n$$

$k$ is the index, $1$ is where it starts and $n$ is where it stops. An infinite series has no last term:

$$\sum_{k=1}^{\infty} a_k = a_1 + a_2 + a_3 + \cdots$$

What an infinite sum means is the subject of 10.02 and 10.03.

The index is a placeholder

$\sum_{k=1}^{4} k^2$ and $\sum_{j=1}^{4} j^2$ are both $1 + 4 + 9 + 16 = 30$. The name of the index doesn't matter; only its values do.

Rules

Constants pull out and sums split, just as for integrals:

$$\sum (c\,a_k + b_k) = c\sum a_k + \sum b_k$$

But a sum of products is not a product of sums: $\sum a_k b_k \ne \left(\sum a_k\right)\left(\sum b_k\right)$.

Formulas worth knowing

$$\sum_{k=1}^{n} 1 = n, \qquad \sum_{k=1}^{n} k = \frac{n(n+1)}{2}, \qquad \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

Shifting the index

The same sum can start anywhere if you adjust the formula. With $j = k - 2$:

$$\sum_{k=3}^{\infty} \frac{1}{(k - 2)^2} = \sum_{j=1}^{\infty} \frac{1}{j^2}$$

Whatever you add to the index, subtract inside the formula (and the other way around). Check by writing out the first term of each.

Splitting off terms

$$\sum_{k=1}^{\infty} a_k = a_1 + a_2 + \sum_{k=3}^{\infty} a_k$$

This is useful when a formula only works from some point on, or when you want to compare two series term by term.

Writing a sum in sigma notation

Find a formula for the $k$-th term, then choose the limits. $1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots = \sum_{k=1}^{\infty} \frac{1}{k^2}$, and $2 + 4 + 6 + \dots + 50 = \sum_{k=1}^{25} 2k$.