Sums and powers of integers

Definition

Sums and powers of integers are fundamental concepts in calculus used to approximate areas under curves. These sums often appear as part of Riemann sums and power series.

5 Must Know Facts For Your Next Test

1. The sum of the first $n$ positive integers is given by $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.
2. The sum of the squares of the first $n$ positive integers is given by $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$.
3. The sum of the cubes of the first $n$ positive integers is given by $\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2$.
4. Riemann sums use partitions and sums of function values to approximate the area under a curve.
5. Power series can represent functions as infinite sums, which are useful for approximating areas in integrals.

Review Questions

• What is the formula for the sum of the first $n$ positive integers?
• How do you express the sum of squares for the first $n$ natural numbers?
• Explain how Riemann sums are used to approximate areas under curves.

Related terms

Riemann Sum: A method for approximating the total area underneath a curve on a graph, usually as a limit of summing up rectangles.

Power Series: An infinite series in which each term is a power function, often used to represent functions within certain intervals.

Definite Integral: Represents the signed area under a curve between two bounds, calculated using limits or antiderivatives.