#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}
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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}
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var('n')
show(limit(1-1/(n+1),n=infinity))
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plot(1-1/(x+1),0,1000)
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#2) sigma((n+1)/(100*n),n,1,infinity) = DNE
f(n)=(n+1)/(100*n)
[f(n).n() for n in [1..10]]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[0.0200000000000000, 0.0150000000000000, 0.0133333333333333, 0.0125000000000000, 0.0120000000000000, 0.0116666666666667, 0.0114285714285714, 0.0112500000000000, 0.0111111111111111, 0.0110000000000000\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[0.0200000000000000, 0.0150000000000000, 0.0133333333333333, 0.0125000000000000, 0.0120000000000000, 0.0116666666666667, 0.0114285714285714, 0.0112500000000000, 0.0111111111111111, 0.0110000000000000\right]
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show(f(x))
show(limit(f(x),x=oo))
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g(x)=(2*x-1)/(9-5*x)
show(g(x))
show(limit(g(x),x=oo))
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#3) d(n)={3,1,4,1,5,...}
#3) sigma(d(n)/10^n)=3+.1+.04+.001+.0005+...=pi
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#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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#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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