Recipe
Evaluate a sum written in sigma notation
Use when: you see \sum with a lower index, an upper index, and a formula for the general term, and need a single number
Goal: the exact number the sum evaluates to
Taught in Sigma notation and finite sums
Quick recipe
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1
Identify the letter used as the index (commonly i, j, or k), the number written below \sum where counting starts, and the number written above where it stops.
Example
For \displaystyle\sum_{j=2}^{6}(2j-3): the index is j, it starts at 2, and it stops at 6.
Why: every later step needs the exact starting and stopping values to know how many terms to compute, and where to begin.
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2
Note the exact expression sitting to the right of \sum, since a single sign or exponent inside it changes every term it produces.
Example
The general term here is 2j-3: double the index and subtract three.
Why: this is the formula every specific term in the next step comes from, so it has to be copied exactly, sign included.
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3
Plug in the starting value, then each following integer, one at a time, up through the ending value, writing down each resulting number.
Example
j=2: 2(2)-3=1; j=3: 2(3)-3=3; j=4: 2(4)-3=5; j=5: 2(5)-3=7; j=6: 2(6)-3=9.
Why: the sum is defined as the total of exactly these terms, no more and no fewer.
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4
Add up the list of numbers from the previous step to get the final value.
Example
1+3+5+7+9=25
⚡ Shortcut
When the general term is linear in the index, each term is a fixed amount more (or less) than the one before it. Skip listing every term by hand: multiply the number of terms by the average of the first and last term. Here there are 5 terms, the first is 1 and the last is 9, so the sum is 5\cdot\dfrac{1+9}{2}=5\cdot5=25, matching the term-by-term total exactly. This only works when consecutive terms change by the same fixed amount every time; it isn't valid for a general term like j^2 or 2^j, where that difference isn't constant.
Why: adding term by term is exactly what the notation means, so it always gives the right value whether or not the shortcut also applies.
Common mistakes
- Treating the index variable as if it appears in the final answer, when it's only a placeholder that disappears once every term is added.
- Starting or stopping at the wrong index value, an off-by-one error, especially when the lower limit isn't 1.
- Mis-substituting into the general term, such as squaring or doubling the index inconsistently from one term to the next.
- Dropping a term at either end of the range, usually the last one.