From dae2b4f5277142de16663d704172da89100be642 Mon Sep 17 00:00:00 2001 From: "Christoph J. Scherr" Date: Wed, 4 Sep 2024 17:03:47 +0200 Subject: [PATCH] numbers --- build/main.pdf | Bin 426142 -> 431688 bytes src/exercise/1.typ | 40 ++++++++++++++++++++++++++++++++++++++++ 2 files changed, 40 insertions(+) diff --git a/build/main.pdf b/build/main.pdf index 8ab6d9cb0cd49085294d566ed0259ecad81c2e24..0fc4e51d03685427fb82c38972930256dbf5d2bf 100644 GIT binary patch delta 103856 zcmZU(byyt1vo0JUgoF^>J;37b1PkuAXmEnNTaXP9TozxP;O?#)Ea>9y?zXt|<#*3L z_uS`x^UplpT~lwpHPbcK)twbUJWx)I4N#SoX5nDvLk0Z&ccG$k2taZikZ~aE_=s4L zulR^KaWD9BM*N7J?2ymjka2K0IVssGfv6%PsOAo4j^AxiQ8^)4gouO?qi@KBac2n# z91u1l1cJIWglohEPh@mRr33Oi$oc~!X#(_Z;d`$C{d7Vp8bw_cGCc|c7erJK882uE zt!W4i;T{!%`~Sg*R1hNIK+p*h1Q2-smkIAU`TnB>GDm^{_}{1){}Bp6;BEQ`-GB%g zB1QPWa3Q#)2>I{%|2HrMgBpRNX^j-&%R2}OJ|a3qQ4jfj6B{+c@dd*FL){FK>%U09 z9xh650UjQR%mZRDr2X;#HWTx|z?hWHJtdTs zOx&yzU~lOm+>eMJ|KqWecSsa(Ir-W7@?wz@n`WL7kB|}go4SyYauF~k#6&2C*x5Px zMNow}I3>g(+Q>*iavp*I)ypUGpI$DW|C0IdmUvSpG7>%l@qgaPe?A}~Ayi6AHbrv> 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zR&;EVLQn{lAA*3OsOGars7a}6J4*R_S$RR-9tf!Ds{x!b;8vhM3+$iy`Qd-3i2~?9Y4Qm|cv0&N za)svaBJv3eApXV@2H}JM?HC4u@%@c243+y&gMZkAq4r4p#~v?gx5Xa`UsaP22IE8R zyZ9FhSF)?~(bemBfCBvde`AhH7yVJF)!Vt*+3gLgBHBbf-e|JAWF9IU)A15fa|Dc876Z{)h)Dq+WpA7j= zbD<0h2>w-LluQuuSB?3h5Y*!RD?2EJ7xp(YPzWFY-^id&IDesqgrE!xApYuLlJhqBa9D^nORCj`u_uA`Jmwd 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 +$