Power series

Question 1

A (complex) power series

Answer 1

A (complex) power series is an expression of the form

SUM (n=0, inf) an(z-zo)^n

where the centre zo and the coefficients an are all fixed complex numbers, and the variable z i complex.

Note: (z-zo)^0 = 1 for all z, even when z = zo

Question 2

Facts about series of complex numbers

Answer 2

SUM (n=0, inf) cn
- The converges depends only on the later terms of the series
- If SUM (n=0, inf) cn converges then cn -> 0 as n-> inf, so the sequence {|cn|} is bounded
- Comparison test
If 0=< |cn| =< dn for all n e |N, and SUM (n=0, inf) dn converges, then so does SUM (n=0, inf) cn
and |SUM(n=0, inf) cn| =< SUM(n=0, inf) dn

Question 3

Radius of convergence

Answer 3

Every power series SUM (n=0, inf) an (z-zo)^n has a radius of convergence, a number in the interval [0, inf] , given by
R = ( lim (n-> inf) sup |an|^1/n)^-1

Question 4

Properties of the radius of convergence

Answer 4

• SUM (n=0, inf) an (z-zo)^n converges if | z - zo| < R
• SUM (n=0, inf) an (z-zo)^n does not converge if |z-zo| > R
• SUM (n=0, inf) an (z-zo)^n may converge for none, some or all z such that |z-zo|=R

Question 5

The ratio test

Answer 5

R = lim(n-> inf) |an| / |an+1| as long as the limit exists

Question 6

The root test

Answer 6

R = lim (n-> inf) 1 / |an|^1/n

Question 7

Stirlings formula

Answer 7

lim (n-> inf) n! / ( (2Pi)^(1/2) e^(-n) n^(n+1/2) ) = 1

Question 8

The algebra and calculus of power series

Answer 8

Suppose that the power series SUM (n=0, inf) an (z-zo)^n and SUM (n=0, inf) bn (z-zo)^n , both converge in B (zo, p) and that c e (C. Then the following power series also converge in B(zo, p):

(1) SUM (n=0, inf) can (z-zo)^n and its sum is c SUM (n=0, inf) an (z-zo)^n
(2) SUM (n=0, inf) (an+bn) (z-zo)^n and its sum is SUM (n=0, inf) an (z-zo)^n + SUM (n=0, inf) bn (z-zo)^n
(3) SUM (n=0, inf) anbn (z-zo)^n and its sum is SUM (n=0, inf) an (z-zo)^n x SUM (n=0, inf) bn (z-zo)^n

Question 9

Differential of a complex power series

Answer 9

Suppose that f(z) = SUM (n=0, inf) an (z-zo)^n  in B(zo, p) and the p>0. Then f'(z) exist in B(zo, p) and 
f'(z) = SUM (n=0, inf) nan (z-zo)^(n-1) = 
SUM (m=0, inf) (m+1)a(m+1) (z-zo)^m 
in B(zo, p)