Refresher - Power Series/ Radius of Convergence
Geometric Series:
The series is said to be a geometric series if each term after the first is a fixed multiple of the term immediately before it. That is, there is a number , called the ratio of the series, such that
, for all
i.e. and
Theorem: If , then the geometric series converges and its sum is .
If and , then the geometric series diverges.
For example the radius of converges of the geometric series is given by , thus the series converges to.
Definition: If the sequence of partial sums of the series diverges to infinity, then we say that the series diverges to infinity and we write .
Other Famous Series:
- The harmonic series is the series
The harmonic series is divergent. - An alternating series is a series where terms alternate signs.
Example:
This converges since for all and . - The series is called the p-series. It converges and diverges if .
- For a given power series there are only three possibilities:
- The series converges only when .
- The series converges for all x.
- There is a positive number R such that the series converges if and diverges if .
The number R in case (c) is called a radius of convergence of the power series. The interval of convergence of a power series is defined to be the interval consisting of all values of x for which the series converges.