List of mathematical series

Mathematical series are important tools in math. They help us find the total of a list of numbers. This list has many formulas for adding up numbers. These formulas work for both lists with a fixed number of terms and lists that go on forever. They are useful for solving hard problems in math and science.

The list explains special symbols used in the formulas. For example, it shows how to write the part of a number after the decimal point. It also introduces important ideas like the Bernoulli polynomials, Euler numbers, and the Riemann zeta function. The list includes symbols for binomial coefficients and the exponential function. These are important parts of many series formulas.

You can use this list with other math tools. These tools include examples of number series and ways to find their totals. They help you understand and calculate these special sums better.

Sums of powers

See Faulhaber's formula.

This section looks at special ways to add up numbers raised to powers. For example, adding up the first few whole numbers looks like this: 1 + 2 + 3 + ... + m, which has a neat formula: m × (m + 1) ÷ 2.

We can also find patterns for adding up squares, cubes, and higher powers. For squares, the sum of the first m squares is m × (m + 1) × (2m + 1) ÷ 6. There are also interesting patterns for adding up fractions where the denominator is a power, like 1 ÷ 1 + 1 ÷ 2 + 1 ÷ 3 + ..., which connect to special numbers like π (pi).

Power series

Mathematical series are sums of numbers that follow special patterns. They can have a fixed number of terms or go on forever. One common type is the power series, which uses powers of a number z.

For example, a simple power series might add up terms like z, z², z³, and so on, up to a certain point. There are easy formulas to find the total. For series that go on forever, special rules apply when the size of z is less than one. These series help solve many problems in math and science!

The article includes links to more details on topics like the geometric series and Multiset.

Trigonometric functions

Sums of sines and cosines appear in Fourier series. Some important series include:

• The sum of the reciprocals of factorials relates to the number e.
• Series involving cosine and sine link to trigonometric functions like sin and cos.

These series help us understand patterns in numbers and are used in many areas of mathematics.