Sigma Notation

Sigma notation is a compact way to write a sum. The symbol $\displaystyle\sum$ (capital Greek sigma) means “add up”, and 0606 uses it to state arithmetic and geometric series concisely.

Reading the notation

$\sum_{r=1}^{n} f(r)$

• $r$ is the counter; it steps up by $1$ each term.
• The number below ($r = 1$) is the starting value.
• The number above ($n$) is the ending value.
• $f(r)$ is the term rule, what to add for each $r$.

So $\displaystyle\sum_{r=1}^{4}(2r + 1)$ means: substitute $r = 1, 2, 3, 4$ into $2r + 1$ and add the results.

A worked example

Evaluate $\displaystyle\sum_{r=1}^{4}(2r + 1)$. $r = 1: 3$; $r = 2: 5$; $r = 3: 7$; $r = 4: 9$. Sum $= 3 + 5 + 7 + 9 = 24$.

This particular sum is an arithmetic series (first term $3$, common difference $2$), so the AP sum formula gives the same answer, useful when the upper limit is large.

Recognising the series type

The term rule tells you which formula to reach for. $\sum(a + (r-1)d)$-style rules are arithmetic; $\sum ar^{r-1}$-style rules are geometric. For a large upper limit, identify the type and use the AP or GP sum formula rather than adding term by term.

Useful starting-value shifts

The counter needn’t start at $1$. $\displaystyle\sum_{r=3}^{6}$ starts at $r = 3$; count the number of terms as (top $-$ bottom $+ 1$), here $6 - 3 + 1 = 4$ terms. Miscounting the number of terms is the usual slip.

Common mistakes

• Miscounting the number of terms (forgetting the "$+1$").
• Starting the substitution at the wrong value of $r$.
• Adding term by term when a sum formula is far quicker for a large limit.

Full topic context: Series notes.