From fbea57919b6ae24b3624fb58939f597d9923a9a5 Mon Sep 17 00:00:00 2001 From: Sven Riwoldt Date: Sat, 29 Jun 2024 08:35:20 +0200 Subject: [PATCH] Exponentials bearbeitet, items erweitert --- exponential.tex | 15 ++++++++++----- upenn.pdf | Bin 344489 -> 344944 bytes 2 files changed, 10 insertions(+), 5 deletions(-) diff --git a/exponential.tex b/exponential.tex index 853c104..516e677 100644 --- a/exponential.tex +++ b/exponential.tex @@ -127,13 +127,18 @@ Each polynomial in the sequence is, in a sense, the best approximation possible \begin{itemize} \item[\textcolor{red}{\tikzcircle{3pt}}] So, how good of an approximation is a polynomial truncation of $e^{\text {? }}$. Use a calculator to compare how close $e$ is to the linear, quadratic, cubic, quartic, and quintic approximations. How many digits of accuracy do you seem to be gaining with each additional term in the series? + + \item[\textcolor{red}{\tikzcircle{3pt}}] Now, do the same thing with $1 / e$ by plugging in $x=-1$ into the series. Do you have the same results? Are you surprised? + + \item[\textcolor{red}{\tikzcircle{3pt}}] Calculate the following sum using what you know: + $$ + \sum_{n=0}^{\infty}(-1)^n \frac{(\ln 3)^n}{n!} + $$ + \end{itemize} %- -%- Now, do the same thing with $1 / e$ by plugging in $x=-1$ into the series. Do you have the same results? Are you surprised? -%- Calculate the following sum using what you know: -%$$ -%\sum_{n=0}^{\infty}(-1)^n \frac{(\ln 3)^n}{n!} -%$$ +%- + %- Write out the first four terms of the following series %$$ %\sum_{n=0}^{\infty}(-1)^n \frac{\pi^{2 n}}{2^n n!} diff --git a/upenn.pdf b/upenn.pdf index 95444e68c8de4040667368e29996581291fa2797..fe93153b4e1fd928e74f1a8533d48d7ce11908ce 100644 GIT binary patch delta 4095 zcmV9~gB9?G6@at>5JUtqFgP`nk>fIdus0c~7``6`il9k1X?rM;tP2E4bI{6K zTc{qk9^LG}?@**{Nzo?hC8q$kMGZd=XNL0)#{?Y^L9aI?JYVcK&t55pD8*JvhIaQz z5QU|+NL!8tp=h^9?<0;S-o}#2=;qzc+n2xJ-rR0K?EbjuSw5e=5)@I58I{5*jk#2R z*jf$)jujI#t&)5}D}`TzngjBKx34!hyC9@US!_7cTwtXsO3Td|ZAKirF5M8!t&!*f zIlvjMD4k+q1m6;E7`Dd128&mUl_Yw6OhMx~MVojr9WPeL({&Di0giw*U_9AjIlok} z*x)#G?ZsLU-XQm>hTrvbpeXh+k0(KY9}8Ac zvx%KDN|Pe(i=-PoYMd(I1at9(NpjCpkK zJ^DJERC}-L+~*aDz1tLOft7rJ?)K`5w=S6A^lzi<%<`yAsz*QC=ZCy&{dmYA%c3Mr z`zA?8MD3rCPHWwJNSkG)7k6R$F7q>oB(L)7;Drx)cl4tyNx}2uU_Duwz2E1~QkQ<5 zv#86O(v5hOS3%;!=>%=oUiW<|)i)^ppZ?xSRt&=^D)yW?GGipVwKgFcj($bLZ@P z^c$aw{&2XM0g1Y@ftJ|NOBwmNYA1nPKwpdqkhu*Ub52Ld--M7gMHVw_7w#^6nQaBN ztYg`rW@F-HL_6z7l*w>KXT*wcq=!*G(r_v1Fn4VkD z7gbRqC^=ML4J+Utrb61FC3AS*~ z!cr_CtUjoGbuBpfA4snt(33$c6Nkk{0k_3Q0eSd1z7xc_1jaz3-;knkOmi?p0vP)jEl+0wHUM!GgGhogIPAt*>ScfGoWfXSH$9i52%mp(tf^kfA6IAyvk` z2DF7CE#1@%>1YD#>``2zrJ%R~D2B8&V-zu@rHbw{LC;Ntf1iv8!b;Y37{1BX$w{?% zBaW=zN{g%t5~1*FJRXb&Hp)udB=V z-_OpMj<-eLo~_r*%L_-_pTGWCzka_yTQ8m8Bvgyj)y3MKoVdj)lmtFDZl@48M$guU zKyMpx3O+Fwe{XDtK5lK$e_Iw*Lql5|=z`H38$|kBAB>r8J!tyk?d9tH{c`O-qrIlZxj>=IRMrbKgZ;hgy%XU#RHeJW3uNyk?=z9POBel||- zF?)ocHV=3F7{(ue|MTDF`L^0O-)vU*kK+$7K>qSc9<<~^M)9B}54olHywQ6<54ji{ zN5BJS2dsX(21ye;WF2m9-SqtFoRcdlSXc9LZ5rNX|mNS)^F>_-v9Jxd!$C-3GK5^gEy@W2+^$ zG8fxnf}e?P2hTQ!Eo}P)+KC<7n74#GJEkyAJ;c`gnb>;zsk-gKxi3cRObYYyEVb%K zX_d$qc>Y|iw(;rHZO;;auiL43QW5joZgv8me~Z_btLyLBy`+cm((n+tfjir#CF z@@1Rkv>(1w!54>_^BBjYjAvSb=D6+2K4P0tjtkCnan^&c7c9#+vOEf$_Z~NZH=N!1!Mh~L- zM`CF~SN9W3r*e;(WTuAG@B_e(?_KaLf1wskVuuc%`!@BD5Rm~_vlk+y3^s&j5+Qu- zhbNsws=mvvy-qSxPTDeV(#|E$a_k3yPamr61KV6q;Mvc_Hq)^mT=y-MyXTY1*p?IZ zxQoJF*dG2cus!zo64Xt_Q%kMtB(3%b8#*G`&>g`>%v^h+Va}XG@utL~2gQI>fARhk z-R{C_@4MX8_2!$s)|B^h7G-Z+6}0QTVPvn^96Dy-`JTcf`}IC%=ANl@c~{Bf?kacT zc`VoCT;3M)xZ6VZRP=6mblmmbUXMr7S}-)8{d#<=B7CU`&oeReN27@HPLBOtMfeel z@WWo1=<`i6*F3P^f?NT*1Ml~=(LPa(+LNqfsLPIt-I7TuyHbO=+Ge$5&GBG|NJT*2(H84U# zMK&`+G&46sLpC-zMlv=wLPjz(MleG%F+N=iFHB`_XLM*FF*7hUlVJoXejoQuTc)E7l!IDx#bX*=0V+XN z++b)%twOb+szD7{O959f8l-^CCAM}{3pVbOBGl7=Yi_&tpfUTep zJi!FTs6ns|44LaM)&wKwPP`Z zUa$}BH+SUMn5)+wf2rdj$jlx6Q}zIpc9^^VT^#bot7NpH0z$7>V zjSG~^UG4bFgBFbB?pbKnA402jd}YNUFc zH}}K4nq?7OrUUB@sBX<*9cTfqme#I{Tm5B8;|f8}+(Q+>f9F8~COuQ&dl>Ht{CRge^0?4?}zk;(hmV7hkEO{^_e@mYIge7-|Wy!cd0ghU7Z&Ynu z2tNgmTk=z%voyJ?ju|j*Y5JKg*T9o!f6CJGm2i=9PoDj3Yzj_w$xk z=mqxYNo)?aU}by&zm6Gj3afKD4bEUWuxu90 zVReY4DLaR)M$MyCU!~~+SO6EnC9nwi&kN4-J?&IF@W&T;Ktl?0mX0i`BgX8C0ul-= zef>?TLQ6lIk}9(F%e_*?mM#@bm00@df>f!0<(gMindL_BOO;!0v0tjfa_^K$Ra)-u z%6wIp`|GaMD$6~da+dq=T3n>vj>wNh6g?MFdL^P_KB9U{MD4#3^}`X3e?+w0is+b* z=uR~iNeoJ&{<{@I)s2z`m5gcyRf|Fe6^a@K)rj&0m50g%RfeJj6@_{P)k7CsrBD?e xC_tb9fdT{y5GX*P0O7LbcQ?mH?wNJ}10bzqJePn$1s)4CGdMH~B_%~qMhdJKt}y@r delta 3660 zcmV-S4zuy_h83xU6@at>5JUquGcuDA0x5rmluK{hFbsh2{uMkbz^v+NTc-jY(r&x0 zO@RT!4yx@Y0%AMy16}|9D9dRb!@z*VqR3AaNl{|32L=xjo5s6cbaS5w;G7z*1?*ly zv4m=4AT^RI#$i{$Zz8Ftn^+4?w!gNIU%x+XpEiGXKR)~{$D4b_fu~e(t-RA%>V$t9 zBTrn9GZnbw%q7+x8eL_tbT>YM)+5-Jf;)W{@2w*k6NHls=yh-VmfwqhB1?#h2zoSNkJ zm=8zWj~TVz72XNq>xs)bu2<9LKq6LYO7f;U*%2AR~y)^P#o*g=oU~ z?!7icK)SN5g1Vo~FJ-Xot*c$V56U;!AA(fc95X+R(X&O^Md5s;zWt;Wq%YgbOOm*H z()e~2?II(&WcO8HjMCc_q-o?k9kX?z9$0hTjX_6lWgWOI{7D-MT>NCCHs zNCL9V4KxZbOl59obZ8(kH8PihEdwck&00;5+eQq%>sQP#kVSI72rP1F(hYJb8YDTS z33AwOjiN=e4mJ*u{`)@K*^RVg?aCUjux}+Y~2NFA)949CU(ID?;`Mp5L{#m4M&1tc3?ev zb2W7?26Gj9F_^0{xfl&L5GQ5=J|In0h8^6F3S*p-OO))Ey2P0pP7zFh0kjP2q`-!{ zcPRo<+=c*SY;4Lf#-LNuWID;^&Mbp;c@su3DvS98(6c0N7}Uz%{4w%*bWXsioDh8J z&?*ypp$Y;05+~OR7AQ~IBb5I+oZe`H(#({H?7h~#wyDTo<>LBF(iwa40Ls)n-aAc6ps~?ADG<}pMD%}<6{`V{r%5>mzP_! 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