diff --git a/build/main.pdf b/build/main.pdf index 8ab6d9c..0fc4e51 100644 Binary files a/build/main.pdf and b/build/main.pdf differ diff --git a/src/exercise/1.typ b/src/exercise/1.typ index e973a0a..9505db1 100644 --- a/src/exercise/1.typ +++ b/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 +$