Real Analysis II:
For the power series, find the radius of convergence R and interval of convergence (IOC)
Kindly find steps/examples for guide in photo II
Transcribed Image Text:Συντήρη
n=1
Transcribed Image Text:EX.
(Boundary
points
=
So (im |an| = lim |+|
ns anti
EX. Find R.O.C. Notice an =
and R=1. check Xx=1:
=
check x = -1 = f(x) = f(-1) = 28 (-1) ² = 1 - 1 + |-|-|-=-=-
C = (-1, 1)
n=0
Div.
16-09. Def f(x) = & an x^² = a + ax+ 9₂x² + 93x²³ +-... is
não
a power
Series
Goal: Find the
domain
Thm: Lef & anx^ be a power Series
if X ERH
no
L
Let R.
=
Then
n=1
an
∞
∞ if x = 0
to
if
Eanx Conv. abs.
Eanx div.
If R =∞, then Eanxn Conv. abs. #XER
If R=0, then Eanx" Conv. only for X ²0
find the
an
of f(x) - (the x's for which som
Eanx Conv.)
I'm suplanta
lim
n=∞an+1
Let X=1. Then & I
n=1 n
กร
if
if Ix > Re R
R.O.C.
lim
336
XnE+
n=1
Let x=-1. Then Ex^²=(-1)^
<= [¹,1)
naln
1x1 <RE R+ and
n
n+l
x "C
lim
A-B
Dive,
7+1
n
Harmonic
R.O.C
Can => 0
n
Conv. by AST lim 1=0, an = 1/
zn
= antl
Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
Expert Solution
Step 1
We will use the fact that a series will be convergence if
