a power series centered at
meaning it is written as
radius of convergence
- there is a radius of convergence
for any power series, possibly infinite, by which the series is absolutely convergent where , and it does not converge when , the case where is uncertain - ratio test: if all coefficients
are nonzero, and some evaluates to some — if is positive or , then that limit is equivalent to the radius of convergence - Taylor’s Formula: a power series
can be differentiated, integrated on the bounds of , the derivatives and integrals will have radius of convergence and to construct the series
linear combinations of power series
When
some power series
geometric series
which converges
exponential series
which converges for all
absolutely convergent
If:
converges, then:
also converges.
This situation is called absolutely convergent.