Sums and powers of integers

Definition

Sums and powers of integers are mathematical operations used to compute series of numbers raised to specific powers. These operations are essential in approximating areas under curves using techniques such as Riemann sums.

5 Must Know Facts For Your Next Test

1. The sum of the first $n$ positive integers is given by the formula $\frac{n(n+1)}{2}$.
2. The sum of the squares of the first $n$ positive integers is given by $\frac{n(n+1)(2n+1)}{6}$.
3. The sum of the cubes of the first $n$ positive integers is $\left(\frac{n(n+1)}{2}\right)^2$.
4. Riemann sums use sums and powers of integers to approximate the area under a curve by dividing it into small rectangles or trapezoids.
5. In integration, these sums help in transitioning from discrete approximations to continuous integrals.

Review Questions

Related terms

Riemann Sum:

A method for approximating the total area under a curve on a graph, typically by dividing it into shapes like rectangles or trapezoids.

Integral:

A fundamental concept in calculus representing the area under a curve; formally defined as the limit of Riemann sums as the number of divisions approaches infinity.

Series:

The sum of terms in a sequence, often expressed using summation notation.