CSH 4.4 MrG 2012.1116 Infinite Sums

2969 days ago by CSH2012

#1) 1/(1*2)+1/(2*3)+1/(3*4)+...+1/(n*(n+1))=1-1/(n+1) #1) 1/1-1/(n+1) 1/(1*2)+1/(2*3)+1/(3*4) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{4}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{4}
1/1-1/2+1/2-1/3+1/3-1/4 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{4}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{4}
var('n') show(limit(1-1/(n+1),n=infinity)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}1
plot(1-1/(x+1),0,1000) 
       
#2) sigma((n+1)/(100*n),n,1,infinity) = DNE f(n)=(n+1)/(100*n) [f(n).n() for n in [1..10]] 
       
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show(f(x)) show(limit(f(x),x=oo)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x + 1}{100 \, x}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{100}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x + 1}{100 \, x}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{100}
g(x)=(2*x-1)/(9-5*x) show(g(x)) show(limit(g(x),x=oo)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x - 1}{5 \, x - 9}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2}{5}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x - 1}{5 \, x - 9}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2}{5}
#3) d(n)={3,1,4,1,5,...} #3) sigma(d(n)/10^n)=3+.1+.04+.001+.0005+...=pi 
       
#4) s = 1+r+r^2+r^3+...+r^n #4) s(n) = a*(1-r^n)/(1-r) #4) s(oo)= a/(1-r),-1<r<1 #4) sigma(2^n,n,1,oo) = 2+4+8+... #4) sigma((1/2)^n,n,1,oo) = 1/2+1/4+1/8+... show(limit(2^x,x=oo)) show(limit(1/2^x,x=oo)) show((1/2)/(1-(1/2))) sum([(1/2)^n for n in [1..100]]).n(digits=100) 
       
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\newcommand{\Bold}[1]{\mathbf{#1}}+\infty
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#5) 1+1/3+1/5+... DNE show(limit(1/(2*x-1),x=oo)) show([1/(2*n-1) for n in [1..10]]) show([1/(2*n-1).n() for n in [1..10]]) show(sum([1/(2*n-1).n() for n in [1..1]])) show(sum([1/(2*n-1).n() for n in [1..10]])) show(sum([1/(2*n-1).n() for n in [1..100]])) show(sum([1/(2*n-1).n() for n in [1..1000]])) show(sum([1/(2*n-1).n() for n in [1..10000]])) show(sum([1/(2*n-1).n() for n in [1..100000]])) 
       
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#6) 1+1/4+1/9+... = pi^/6 show(limit(1/(x^2),x=oo)) show([1/(n^2) for n in [1..10]]) show([1/(n^2).n() for n in [1..10]]) show(sum([1/(n^2).n() for n in [1..1]])) show(sum([1/(n^2).n() for n in [1..10]])) show(sum([1/(n^2).n() for n in [1..100]])) show(sum([1/(n^2).n() for n in [1..1000]])) show(sum([1/(n^2).n() for n in [1..10000]])) show(sum([1/(n^2).n() for n in [1..100000]])) show((pi^2/6).n()) 
       
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