MATH2310 – DE Lecture 6
Power series:
Some well-known power series include:
A power series centred at x=a is given by:
∞
∑ c k ( x−a)k
k= K
For any fixed x , it is an infinite series that may or may not converge.
The radius of convergence, R ≥ 0,k is the number such that for |x−a|< R , the series
converges, and for |x−a|> R , the series diverges.
The ratio test for convergence is:
c k +1
|x−a| lim =|x−a| L
k →∞ ck
If L<1, the series is convergent,
If L>1, the series is divergent,
If L=1, the test was unclear.
The radius of convergence does not say anything about what happens at |x−a|=R .
The interval of convergence is the interval of those x ϵ R such that the series
Power series:
Some well-known power series include:
A power series centred at x=a is given by:
∞
∑ c k ( x−a)k
k= K
For any fixed x , it is an infinite series that may or may not converge.
The radius of convergence, R ≥ 0,k is the number such that for |x−a|< R , the series
converges, and for |x−a|> R , the series diverges.
The ratio test for convergence is:
c k +1
|x−a| lim =|x−a| L
k →∞ ck
If L<1, the series is convergent,
If L>1, the series is divergent,
If L=1, the test was unclear.
The radius of convergence does not say anything about what happens at |x−a|=R .
The interval of convergence is the interval of those x ϵ R such that the series
