Capital-sigma notation

Capital-sigma notation

Mathematical notation uses a symbol that compactly represents summation of many similar terms: the summation symbol, ${\textstyle \sum }$, an enlarged form of the upright capital Greek letter sigma. This is defined as $${\displaystyle \sum _{i\mathop {=} m}^{n}a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n}}$$ where i is the "index of summation" or "dummy variable", a_i is an indexed variable representing each term of the sum; m is the "lower bound of summation", and n is the "upper bound of summation". The "i = m" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by one for each successive term, stopping when i = n. This is read as "sum of a_i, from i = m to n". However, some notations may include the index at the upper bound of summation, or omit the index at the lower bound as in ${\textstyle \sum _{i=m}^{i=n}a_{i}}$ or ${\textstyle \sum _{m}^{n}a_{i}}$, respectively. There are sigma notation variants where the range of bounds is omitted, which denotes the dummy variable only, like ${\textstyle \sum _{i}a_{i}}$. Here is an example showing the summation of squares: $${\displaystyle \sum _{i=3}^{6}i^{2}=3^{2}+4^{2}+5^{2}+6^{2}=86.}$$ In general, while any variable can be used as the index of summation (provided that no ambiguity is incurred), some of the most common ones include letters such as ${\displaystyle i}$, ${\displaystyle j}$, ${\displaystyle k}$, and ${\displaystyle n}$; the latter is also often used for the upper bound of a summation. Alternatively, the index and bounds of summation are sometimes omitted from the definition of summation if the context is sufficiently clear. This applies particularly when the index runs from 1 to n. For example, one might write that ${\textstyle \sum a_{i}=\sum _{i=1}^{n}a_{i}}$.