Mach 9.5 Series

195 days ago by mpaul

\sum Notation

A series is the sum of a sequence.

If \{a_n\} = \{a_1, a_2, a_3, ..., a_n\}, then

\sum _{k = 1}^{n} a_k = a_1 + a_2 + a_3 + ... + a_n.
a(n) = 2*n-1 [a(k) for k in [1..10]]; sum(_) 
       

\sum_{k=1}^{5} 3k
[3*k for k in [1..5]]; sum(_) 
       

\sum_{k=5}^{8} k^2
[k^2 for k in [5..8]]; sum(_) 
       

\sum _{n=0}^{12} \cos(n\pi)
[cos(n*pi) for n in [0..12]]; sum(_) 
       

\sum _{n=1}^{\infty} \sin(n\pi)
[sin(n*pi) for n in [0..100]]; sum(_) 
       

\sum_{k=1}^{\infty} \frac{3}{10^k}
[3/10^k for k in [1..20]]; sum(_).n() 
       

Arithmetic Series
first,last,num = var('a_1,a_n,n') @interact def arithmetic(a = 0, d = 1, n = 10): seq = [a+k*d for k in range(n)] show(seq) show(num/2*(first+last)) print '1/2*',n,'(',a,'+',a+(n-1)*d,') = ' show(sum(seq)) 
       

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a_1, d = var('a_1 d') a(i) = a_1 + (i-1)*d S(n) = (a(1)+a(n))*n/2 
       
S(10) 
       
 
       

Geometric Series
first,ratio,num = var('g,r,n') @interact def geometric(g = 1, r = 2, n = 10): seq = [g*r^k for k in range(n)] show(seq) show(first*(ratio^num-1)/(ratio-1)) print g,'(',r^n,'-',1,')/(',r,'-',1,') = ' show(sum(seq)) 
       

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Partial Sums