Power Series Flashcards
What is the form of a power series
cn(x - a)n
What are the components of a power series
Form:cn(x - a)n
cn -> coefficients of the power series
a -> centre of the power series
Definition: interval of convergence (IC)
All the values of x for with a power series in convergent
Definition: radius of converge (R)
Distance from the centre (a) to either boundary of the interval of convergence
What methods can be used to find the con/div (and therefore interval and radius of convergence) of power series
- geometric series test
- ratio test
Which interval/radius- finding test requires the end points be checked
Ratio test
What does it mean when power series only converges when x = a
IC: x = a
R = 0
What does it mean when the power series is convergent for all x
IC: (-infinity, infinity)
R = infinity
What does the series [sum from 0 to infinity] xn represent
1/1-x, where |x|< 1
Do the upper/lower bounds of the series change when you differentiate a power series
Yes! (if the first term is a constant, otherwise no)
The lower bound is one greater (n = 1 instead of n = 0)
Do the upper/lower bounds of the series change when you differentiate a power series
No
What do we know if f(x) = [series from 0 to infinity] cn(x - a)n with radius of convergence R > 0 or R = infinity
Theorem 2 of power series
fâ(x) = [series from 1 to infinity] cn(x - a)n - 1
and
integral of f(x)dx = C + [series from 0 to infinity] cn/n+1 * ((x - a)n + 1)
Both still have radius of convergence R (same as before)
Does integrating or differentiating a power series change the radius of convergence
No
