Summation notation

Definition

Summation notation, often represented by the Greek letter sigma ($\Sigma$), is a concise way to express the sum of a sequence of numbers. It is commonly used to approximate areas under curves and in various mathematical analyses.

5 Must Know Facts For Your Next Test

1. Summation notation uses an index variable (usually $i$, $j$, or $k$) that runs from a lower bound to an upper bound.
2. The general form is $\sum_{i=a}^{b} f(i)$, where $a$ is the lower limit, $b$ is the upper limit, and $f(i)$ is the function being summed.
3. In calculus, summation notation can be used to represent Riemann sums which approximate integrals.
4. Summation properties include linearity: $\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$ and constant multiplication: $\sum_{i=1}^{n} c \cdot a_i = c \cdot \sum_{i=1}^{n} a_i$.
5. It is important for understanding definite integrals in Calculus II as they are the limits of Riemann sums.

Review Questions

• What does the index variable in summation notation represent?
• How would you write the sum of squares from 1 to n using summation notation?
• What are two key properties of summation notation that simplify calculations?

Related terms

Riemann Sum: A method for approximating the total area under a curve by dividing it into smaller subintervals and summing up areas of rectangles or trapezoids.

Definite Integral: The evaluation of an integral with upper and lower bounds; it represents the net area under a curve over an interval.

Index Variable: A variable used in summation notation that indicates each term's position within the sequence.