Houjun Liu

power series

a power series centered at a is defined with cnR, whereby:

f(x)=n=0cn(xa)n

meaning it is written as c0+c1(xa)+c2(xa)2+c3(xa)3+

radius of convergence

  • there is a radius of convergence R0 for any power series, possibly infinite, by which the series is absolutely convergent where |xa|<R, and it does not converge when |xa|>R , the case where |xa|=R is uncertain
  • ratio test: if all coefficients cn are nonzero, and some limn|cncn+1| evaluates to some c — if c is positive or +, then that limit is equivalent to the radius of convergence
  • Taylor’s Formula: a power series f(x) can be differentiated, integrated on the bounds of (aR,a+R), the derivatives and integrals will have radius of convergence R and cn=f(n)(a)n! to construct the series

linear combinations of power series

When n=0an and n=0bn are both convergent, linear combinations of them can be described in the usual fashion:

c1n=0an+c2n=0bn=n=0c1an+c2bn

some power series

geometric series

1+r+r2+r3+=n=0rn=11r

which converges 1<r<1, and diverges otherwise.

exponential series

1+x+x22!+x33!+=n=0xnn!=ex

which converges for all xR.

absolutely convergent

If:

n=0|an|

converges, then:

n=0an

also converges.

This situation is called absolutely convergent.