diff --git a/build/main.pdf b/build/main.pdf index 53c25d2..8ab6d9c 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 new file mode 100644 index 0000000..e973a0a --- /dev/null +++ b/src/exercise/1.typ @@ -0,0 +1,53 @@ +== Exercise Sheet 1 - Basics + +=== Exercise 1 @Exercise[1, 1] + +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + Determine whether the following sequences converge. Calculate the limit in case of convergence. + + #set enum(numbering: "(a)") + + $ a_n = (2024 (1+n+n^2))/(n(n+2023)) $ + + $ a_n = sqrt(n^2+n dot b_1+b_2)-n; quad b_1,b_2 in RR $ + + $ a_n = (n^4-2)/(n^2+4) + (n^3(3-n^2))/(n^3+1) $ +]) + +#set enum(numbering: "(a)") ++ $ + a_n &= (2024 (1+n+n^2))/(n(n+2023)) \ + &= ((2024 (1+n+n^2))/n^2)/(1+2023/n) \ + &= (2024 (1/n^2+1/n+1/))/(1+2023/n) \ + &= (2024/n^2+2024/n+2024)/(1+2023/n) \ + + => lim_(n -> infinity) a_n &= lim_(n -> infinity) (2024/n^2+2024/n+2024)/(1+2023/n) \ + &= lim_(n -> infinity) 2024/1 = 2024 checkmark \ +$ + ++ $ + a_n &= sqrt(n^2+n dot b_1+b_2)-n; quad b_1,b_2 in RR \ + &= ((sqrt(n^2+n dot b_1+b_2)-n) (sqrt(n^2+n dot b_1+b_2)+n))/(sqrt(n^2+n dot b_1+b_2)+n) \ + &= (n^2+n dot b_1+b_2 - n^2)/(sqrt(n^2+n dot b_1+b_2)+n) \ + &= (n dot b_1+b_2)/(sqrt(n^2+n dot b_1+b_2)+n) \ + &= (b_1+b_2/n)/((sqrt(n^2+n dot b_1+b_2)+n)/n) \ + &= (b_1+b_2/n)/((sqrt(n^2+n dot b_1+b_2))/n+1) \ + &= (b_1+b_2/n)/((sqrt(n^2+n dot b_1+b_2))/sqrt(n^2)+1) \ + &= (b_1+b_2/n)/(sqrt(1+b_1/n+b_2/(n^2))+1) \ + + => lim_(n -> infinity) a_n &= lim_(n -> infinity) (b_1+b_2/n)/sqrt(1+b_1/n+b_2/(n^2))+1 \ + &= b_1/(sqrt(1) + 1) = b_1/2 checkmark +$ + ++ $ + a_n &= (n^4-2)/(n^2+4) + (n^3(3-n^2)) / (n^3+1) \ + &= ((n^4-2)(n^3+1) + (n^3(3-n^2))(n^2+4)) / ((n^2+4)(n^3+1)) \ + &= (n^7+n^4-2n^3-2 + 3n^5+12n^3-n^7-4n^5) / ((n^2+4)(n^3+1)) \ + &= (-n^5+n^4+10n^3-2) / (n^5+n^2+4n^3+4) \ + &= (-1+1/n+10/(n^2)-2/(n^5)) / (1+1/(n^3)+4/(n^2)+4/(n^5)) \ + + => lim_(n -> infinity) a_n + &= 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 +$ diff --git a/src/exercise/index.typ b/src/exercise/index.typ index e69de29..aa9b3b3 100644 --- a/src/exercise/index.typ +++ b/src/exercise/index.typ @@ -0,0 +1,2 @@ + +#include "1.typ" diff --git a/src/vorlesungen/1.typ b/src/vorlesungen/1.typ index c896b3b..f8a6250 100644 --- a/src/vorlesungen/1.typ +++ b/src/vorlesungen/1.typ @@ -330,7 +330,7 @@ $ An dieser Stelle stimmt die Lösung in @Vorlesung leider wieder nicht. Ich habe das Ergebnis wie in maschinell überprüfen lassen, und es war -korrekt. Lauf @Vorlesung wäre das Ergebnis: +korrekt. Laut @Vorlesung wäre das Ergebnis: $ A = integral x sin(x) d x = x(-cos(x)) - integral sin(x) d x = cos(x)(1-x) " "#emoji.crossmark $