Textbook
For integers $m\le n$, and numbers $a_i$, a finite sum is $\sum_{i=m}^na_i:=a_m+a_{m+1}+\cdots+a_n$. Both bounds are included, and index $i$ selects each term. A finite index set may also be used: $\sum_{i\in I}a_i$. The empty sum is zero.
A bound index may be renamed if no name conflict arises. In $\sum_{i=1}^na_{ij}$, the index $i$ disappears after summation, while $j$ remains free: each column has its own result. A subscript labels an element; it is not a multiplier. The mathematical sum specifies a result, while machine evaluation order and rounding are separate questions. Infinite sums require convergence analysis.
Bounds from $m$ through $n$ include $n-m+1$ terms. If the expression being summed is a constant $c$, the result is therefore $(n-m+1)c$. Parentheses define the scope: $\sum_{i=1}^n(a_i+b_i)$ adds both values for every index. A number written outside an already completed sum is added once rather than automatically on every pass through the indices.
Definitions and further reading: Mathematics for Computer Science.
In plain language
A capital sigma totals selected receipt lines: the bounds specify their indices, while the expression inside specifies each contribution. Index names $i$ and $j$ do not affect the total, just as renaming a cashier does not change a purchase. Line 3 need not cost three roubles; an index selects a term rather than replacing it. An empty receipt totals zero, but missing bounds leave even a diligent cashier unsure when to stop.
Example
The example accumulates the sum of the original illustrative measurements $x_1=2$, $x_2=4$, and $x_3=6$. After the third step the bound index no longer appears in the result 12.
Finite Sums: Reading the Summation Sign
Index 1
The initial zero increases by the first value, 2.
Step 1 of 3: Index 1
Read all steps
Index 1
The initial zero increases by the first value, 2.
- State
- 0current+2current2current
Index 2
Adding 4 increases the accumulated 2.
- State
- 2current+4current6current
Index 3
The last term, 6, brings the total to 12.
- State
- 6current+6current12current
The second result, 6, already includes the first two measurements. The final step therefore adds only the last 6 to obtain 12, without counting earlier terms again. The complete sum expands as $2+4+6$ for a check. The result 12 need not equal the last value 6: the index has finished its work, but the earlier contributions remain in the accumulated answer.
Related concepts: Finite products.