Power Series
A power series is an infinite series of the form
∑
n0
cnxnc0c1xc2x2c3x3cnxn
where xis a variable is called a power series. More generally, series of the form
∑
n0
cnx−anc0c1x−ac2x−a2cnx−an
is called a power series centered at aor the power series about a,whereais a constant.
Note: to simplify notation we let x−a01 even when xa. For a power series centered at
athere are only THREE possibilities:
(1) The series converges only when xa.
(2) The series converges for all x.
(3) There is a positive real number Rsuch that the series converges if |x−a|Rand
diverges if |x−a|R.
The number Ris called the radius of convergence of the power series. In case (1), R0.
In case (2), R. The set of all values of xfor which the power series converges is called the
interval of convergence of the power series.
We use the Ratio Test to find the radius of convergence of a power series. For a power
series centered about awith a finite radius of convergence R0, we must determine whether
or not the series converges at the endpoints of the interval of convergence (i.e. at a−R&
aR).
Find the interval of convergence for the following power series.
1. ∑
n0
n!xn2. ∑
n0
−1nx2n1
2n1!3. ∑
n1
xn
n4. ∑
n0
−1nx1n
2n5. ∑
n1
x−2n
n2
Term-by-term differentiation and integration of functions defined by power series:
Let the power series ∑
n0
cnx−anhave radius of convergence R0. Then the function
fx∑
n0
cnx−anc0c1x−ac2x−a2c3x−a3
is differentiable (and therefore continuous) on the interval a−R,aRand
(a)f′x∑
n0
ncnx−an−1c12c2x−a3c3x−a2
(b)fxdx C∑
n0
cnx−an1
n1Cc0x−ac1x−a2
2c2x−a3
3
The series in (a) and (b) have radius of convergence R. Convergence or divergence at the
endpoints of the interval of convergence may differ from the original power series.
Find f′xand fxdx and their corresponding intervals of convergence if fx∑
n1
xn
n.
If gx∑
n1
xn
n!find g′x. Do you recognize this function?