Writing out a long sum is tedious. Sigma notation packs the whole instruction
into one symbol — the Greek capital sigma \sum, which simply means
"add these up".
\sum_{k=1}^{5} k \;=\; 1 + 2 + 3 + 4 + 5 \;=\; 15.
You read it as a recipe. The letter k is the
index (a counter). It starts at the number below the sigma — the
lower limit — climbs by one each time, and stops at the number on top — the
upper limit. The expression after the sigma is the
general term: substitute each value of k into it
and add all the results.
Reading the recipe
The general term can be any rule. Whatever it is, you substitute
k = 1, 2, 3, \dots up to the top, then total the terms:
\sum_{k=1}^{4} 2k \;=\; 2 + 4 + 6 + 8 \;=\; 20,
\sum_{k=1}^{3} k^2 \;=\; 1 + 4 + 9 \;=\; 14.
The lower limit need not be 1. In
\sum_{k=2}^{4} k^2 = 4 + 9 + 16 = 29 the counter runs from
2 to 4, so there are three terms. This is
exactly how an
arithmetic or geometric series
is written compactly — the general term is just the sequence's
nth-term rule.
In \displaystyle\sum_{k=1}^{n} a_k:
- \sum means add the terms;
- k is the index, running from the bottom value
up to the top value, one step at a time;
- a_k is the general term — substitute each
k into it;
- the number of terms is (\text{upper}) - (\text{lower}) + 1.