Summary Power Series | Calculus II Notes

3.5 Power Series
series of the form
A Ao+A,(X-C) Az(X-c) +Az(x-C)3+... An(x-C)"
+ =


is called a power
series in (x-c) or a power series centered on c where An are called coefficients ofthe power series



power series centered on C0 =

reduce to Ao+A,x+AcX +AgX+... cAnX" -




example: n!
so
this is a power series centered on c0=

and An= I!

CoefficientAn are
typically given fixed numbers butX is thoughtof as a variable, meaning each power
series is more ofa family/groupofseries, a differentseries for every value of X



one possible value of XisX c =




·An(X-C)" x c =
noAn(-c)"
=




=

Ho 0 + 0+Ot... +




which As
converges to

with this, we know that a power series converges when x c, and
=

can use other convergence tests

to find for which other values of X the series converges


using the ratio test
can An(X-c)
=




eim Ant1=lim Anti(X-C)n+1
n D
- n 0
=



an An(X-C)"
=ein Ant. X-C
n 0 -



An

=X- C line An+ 1
n 0
=

An



Art=
if im Ao
n N
=




tells
the ratio test us noAn(X-c)" converges when A.x-CC1 <IX-C 'A
A.x-C>1 A
diverges when x-C

When the limitexists
-
I

:R A lim
= =

Anti
n 0 =


An

R is called the radius ofconvergence


ratiotest tells neoAn(X-c" converges
Ant andthe
Anti us that
ifhim 0 then
in 1x-C 0 for every
=
=




n =0 An


for every number x

series has an infinite radius
the of convergence

if m Anti +00
=




(diverges to infinity) then him
n D +
Anti x -c
+
=
20 for every x/C and the ratio test
An An

tells us that An(X-C)" diverges for every number x =c

the series has radius convergence zero
of




if
Ant
does not approach a limit as, then we learn
nothing