List of mathematical series

Mathematical series are important tools in math that help us find the sum of sequences of numbers. This list offers many formulas for adding up both finite (with a set number of terms) and infinite (going on forever) series. These formulas can be very useful when solving complex problems in many areas of mathematics and science.

The list explains special notations used in these formulas. For example, it shows how the fractional part of a number is written, and introduces important functions like the Bernoulli polynomials, Euler numbers, and the Riemann zeta function. It also includes symbols for binomial coefficients and the exponential function, which are key parts of many series formulas.

You can use this list together with other math tools, like the examples of numerical series and methods for summation, to better understand and calculate these special sums.

Sums of powers

See Faulhaber's formula.

This section explores 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 fascinating 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 following specific patterns. They can be finite (with a set number of terms) or infinite (going on forever). One common type is the power series, which involves terms like powers of a number z.

For example, a simple finite power series might add up terms like z, z², z³, and so on, up to a certain point. There are neat formulas to find the total quickly. For infinite series, special rules apply when the absolute value 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.