numbers

src/exercise/1.typ

@@ -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 = 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
+$