Real Analysis II:
Kindly solve problem in photo 1 with notes from photo 2
Transcribed Image Text:5. Find a series 1 a, which converges conditionally such that Σ;=1 a²
diverges.
Transcribed Image Text:{a}
09. AST. I fant in
numbers and
Defi Eaj
ex. 2.0
Conv. by
A-S-T
ex.
|a₁ ≤
(
8 W.
&a
fact
a
lim an o
is abs Conv.
abs. Conv..
because
J+1
Study 3/5 = 1/2 1/²
"1
is
E
B₁Aj
every Eaj
decreasing seg of positive
then & (-1)+
NOT
Thim: (Absolute Convergence Test)
If Assume
If Eaj Conv. abs, then 2(a) Conv.
Eaj
Conv. abs.
[Since |aj| = |aj| t; = 1, ...
j
we Ɛaj
get Eaj Conv.
80
if 2 lail Conv...
yes
abs. Conv.
an Converges-
Conv. (P-Sehes
p=2
X EA CON.
(Harmonic, Siv)
Problem
Bj
Elajh Conv.
aux
Coution the Converse is false
Def if Eaj Conv. but does not
Conv. abs then Eag Convergo Condition
22
either diverges Conv. abs,
от
Conv. Conditi
EGI
A
= 1 = 1/2 + 1/3 - 1/14 + 1/5 - 1/6 + 1/ 7- / 8 + 1/ 7- / 10 + 1/11 - 1/12 + 1/3 - 1/12/
#½-1⁄2-1⁄4-16-18-1/10 - 1/12 - 1/14
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.
