Sigma Notation and Evaluating Sums Cheat Sheet

Key Facts

• In $\sum_{i=1}^{n} a_i$, the index is $i$, the lower limit is $1$, the upper limit is $n$, and $a_i$ is the expression being added.
• To expand a sum, substitute each integer value of the index from the lower limit to the upper limit into the expression.
• The constant multiple rule is $\sum_{i=m}^{n} c a_i = c\sum_{i=m}^{n} a_i$.
• The sum rule is $\sum_{i=m}^{n} (a_i + b_i) = \sum_{i=m}^{n} a_i + \sum_{i=m}^{n} b_i$.
• The constant sum formula is $\sum_{i=1}^{n} c = cn$.
• The first $n$ positive integers add to $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.
• The first $n$ squares add to $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$.
• A finite geometric sum is $\sum_{i=0}^{n} ar^i = \frac{a(1-r^{n+1})}{1-r}$ when $r \neq 1$.

Vocabulary

Sigma notation

Sigma notation uses the symbol $\sum$ to show that a sequence of terms should be added.

Index of summation

The index of summation is the variable, such as $i$ or $k$, that changes value in each term of the sum.

Lower limit

The lower limit is the starting value of the index in a sum, such as $1$ in $\sum_{i=1}^{n} a_i$.

Upper limit

The upper limit is the ending value of the index in a sum, such as $n$ in $\sum_{i=1}^{n} a_i$.

Arithmetic series

An arithmetic series is a sum of terms with a constant difference, often evaluated by $S_n = \frac{n}{2}(a_1+a_n)$.

Geometric series

A geometric series is a sum of terms with a constant ratio, often written as $\sum_{i=0}^{n} ar^i$.