Power series

A power series centered at c is defined as the infinite sum

a_nxⁿ is called the n^th term of the power series.

a_n is called the n^th coefficient of the power series.

Notice that we are adding up terms with increasing powers of (x - c), hence the name power series.

Power series are used to approximate functions that are difficult to calculate exactly, such as tan^-1(x) and sin(x), using an infinite series of polynomials. Power series are often used to approximate important quantities and functions such as π, e, and , an important function in statistics.

In real life, we cannot add an infinite number of terms together since any computer can only hold a certain amount of memory. Therefore, we approximate a power series using the th partial sum of a power series, denoted S_n(x). To produce the n^th partial sum, we cut off the infinite series after the n^th term, getting rid of all terms with powers of (x - c) higher than n. We only keep the first n + 1 terms of the power series (remember that we start from the 0^th term which is f(c)). The n^th partial sum is defined as:

If x = x₀ and

exists, then the power series is said to converge at x₀. Otherwise, the power series diverges at x₀.

In cases where c = 0, the infinite sum is

The domain of f, often called the interval of convergence (IOC), is the set of all x-values such that the power series converges.

• For a given power series, it can be proven that either the IOC = (-∞,∞), meaning that the series converges for all x, or there exists a finite non-negative number R ≥ 0, called the radius of convergence (ROC), such that the series converges whenever |x - c| < R and diverges whenever |x - c| > R.
• By convention, when the IOC of f is (-∞, ∞), the ROC is R = +∞.
• When R = 0, the IOC consists only of c.
• We can find the IOC by first finding the ROC with the ratio or root test, and then testing the endpoints c ± R with some other test like integral, comparison, alternating series, p-series, etc.
• If the ROC is a finite number R, the IOC will always be of one of the following forms:
  • [c - R, c + R]
  • (c - R, c + R]
  • [c - R, c + R)
  • (c - R, c + R)

Geometric series

A geometric series is one of the most important examples of a power series. In a geometric series, all the a's are the same and c = 0:

The n^th partial sum of f is given by:

S_n(x) = a + ax + ax² + ax³ + ax⁴ + ... + axⁿ

We can also write S_n as:

This can be checked by multiplying both expressions for S_n(x) by x - 1 and realizing that all powers of x cancel out except the 0^th and n+1^th:

Therefore:

If |x| < 1, then because each time we increase n, we are multiplying by x which has size smaller than 1, and

If |x| > 1, then and

and the series diverges.

If x = -1, then

S_n(x) alternates between a and 0 so does not exist and the series diverges.

If x = 1, then

We have just proven that

if and only if |x| < 1, i.e., x is in the interval (-1, 1). Therefore, the geometric series has ROC 1 centered at 0 and IOC = (-1, 1).