Refresher - Power Series/ Radius of Convergence

Geometric Series:

The series n=0an 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 r, called the ratio of the series, such that

an+1= ran,  for all  n0

i.e.  an=aorn  and  n=0an=a0+ra0+r2a0+r3a0+

Theorem: If |r|<1, then the geometric series n=0an converges and its sum is S=n=0a0rn=a01-r.
If |r|>1 and a00, then the geometric series diverges.
For example the radius of converges of the geometric series n=0(12)2 is given by r=12<1, thus the series converges tor=11(12)=2.

Definition: If the sequence of partial sums of the series n=0andiverges to infinity, then we say that the series diverges to infinity and we write n=0an= .

 

Other Famous Series:

  • The harmonic series is the series
    1+12+13+14+15=n=11n. The harmonic series is divergent.
  • An alternating series is a series where terms alternate signs.
    Example:
    1-12+13-14+15-=n=1(-1)n+11n. This converges since an+1an for all n and limnan=0.
  • The seriesn=1pn is called the p-series. It convergesifp>1 and diverges if p1.
  • For a given power series n=0cn(x-a)n there are only three possibilities:
    1. The series converges only when x=a.
    2. The series converges for all x.
    3. There is a positive number R such that the series converges if |x-a|<R and diverges if |x-a|>R.
      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.