Power series
A power series centered at c is defined as the infinite sum
= | |||
= |
a_{n}x^{n} 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 = x_{0} and
exists, then the power series is said to converag eat x_{0}. Otherwise, the power series diverges at x_{0}.
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:
f(x) | = | a + ax + ax^{2} + ax^{3} + ax^{4} + ... | |
= |
The n^{th} partial sum of f is given by:
S_{n}(x) = a + ax + ax^{2} + ax^{3} + ax^{4} + ... + ax^{n}
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
= | |||
= | a if n is even; 0 if n is odd |
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).