Real Analysis II:Kindly find steps/examples for guide in photo II 1. For each of the following power series, find the radius of convergenceR and the interval of convergence (IOC) (prove your answers).(a)(b)8n=13ηη-xn∞Σ()"n=1 EX.(Boundarypoints=So (im |an| = lim |+|ns antiEX. 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=0Div.16-09. Def f(x) = & an x^² = a + ax+ 9₂x² + 93x²³ +-... isnãoa powerSeriesGoal: Find thedomainThm: Lef & anx^ be a power Seriesif X ERHnoLLet R.=Thenn=1an∞∞ if x = 0toifEanx Conv. abs.Eanx div.If R =∞, then Eanxn Conv. abs. #XERIf R=0, then Eanx" Conv. only for X ²0find theanof f(x) - (the x's for which somEanx Conv.)I'm suplantalimn=∞an+1Let X=1. Then & In=1 nกรifif Ix > Re RR.O.C.lim336XnE+n=1Let x=-1. Then Ex^²=(-1)^<= [¹,1)naln1x1 0nConv. by AST lim 1=0, an = 1/zn= antl

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Answered: 1. For each of the following power…

1. For each of the following power series, find the radius of convergence R and the interval of convergence (IOC) (prove your answers). (a) (b) 8 n=1 3η η -xn ∞ Σ()" n=1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Real Analysis II: Kindly find steps/examples for guide in photo II

1. For each of the following power series, find the radius of convergence
R and the interval of convergence (IOC) (prove your answers).
(a)
(b)
8
n=1
3η
η
-xn
∞
Σ()"
n=1

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.

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