This commit is contained in:
Christoph J. Scherr 2024-09-04 17:03:47 +02:00
parent 4681ed7a2b
commit dae2b4f527
2 changed files with 40 additions and 0 deletions

Binary file not shown.

View File

@ -51,3 +51,43 @@ $
&= lim_(n -> infinity) (-1+1/n+10/(n^2)-2/(n^5)) / (1+1/(n^3)+4/(n^2)+4/(n^5)) \ &= lim_(n -> infinity) (-1+1/n+10/(n^2)-2/(n^5)) / (1+1/(n^3)+4/(n^2)+4/(n^5)) \
&= lim_(n -> infinity) -1/1 = 1 checkmark &= lim_(n -> infinity) -1/1 = 1 checkmark
$ $
#pagebreak()
=== Exercise 2 @Exercise[1, 2]
#block(
fill: luma(230),
inset: 8pt,
radius: 4pt,
[
Examine whether the following series converge or diverge.
#set enum(numbering: "(a)")
+ $ A = sum_(n=1)^infinity (2^n n!)/(n^n) $
+ $ A = sum_(n=1)^infinity 1/(n^n) $
])
#set enum(numbering: "(a)")
+ $
A &= sum_(n=1)^infinity (2^n n!)/(n^n) \
a_n &= (2^n n!)/(n^n) #text("Ratio Test") \
=> lim_(n -> infinity) a_n
&= lim_(n -> infinity) abs((a_(n+1))/(a_n)) \
&= lim_(n -> infinity) abs( ( (2^(n+1) (n+1)!)/((n+1)^(n+1)) )/( (2^n n!)/(n^n) )) \
&= lim_(n -> infinity) abs( ( 2^(n+1) dot (n+1)! dot n^n )/( 2^n dot n! dot (n+1)^(n+1) )) \
&= lim_(n -> infinity) abs( ( 2 dot (n+1)! dot n^n )/( n! dot (n+1)^(n+1) )) \
&= lim_(n -> infinity) abs( ( 2 dot (n+1) dot n^n )/( (n+1)^(n+1) )) \
&= lim_(n -> infinity) abs( ( 2 dot n^n )/( (n+1)^n )) \
&= lim_(n -> infinity) abs( 2 dot ( n^n )/( (n+1)^n )) \
// &= lim_(n -> infinity) abs( 2 dot (n/(n+1))^n ) \
// &= lim_(n -> infinity) 2 dot (n/(n+1))^n \
&= lim_(n -> infinity) 2 dot (1/(1+1/n))^n \
&= 2 dot e > 1 => a_n thin #text("diverges") checkmark
$
+ $
a_n &= 1/(n^n) #text("Root Test") \
root(n, abs(1/(n^n))) &= 1/n -> 0 < 1 \
&=> a_n #text(" diverges") checkmark
$