A power series centered at c is defined as the infinite sum
anxn is called the nth term of the power series.
an is called the nth 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 Sn(x). To produce the nth partial sum, we cut off the infinite series after the nth 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 0th term which is f(c)). The nth 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)
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:
|f(x)||=||a + ax + ax2 + ax3 + ax4 + ...|
The nth partial sum of f is given by:
Sn(x) = a + ax + ax2 + ax3 + ax4 + ... + axn
We can also write Sn as:
This can be checked by multiplying both expressions for Sn(x) by x - 1 and realizing that all powers of x cancel out except the 0th and n+1th:
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
|=||a if n is even; 0 if n is odd|
Sn(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).