Re: Series Programs
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Re: Series Programs
Look up Fibonacci, a 13th century Italian mathematician from Pisa as well as
Orthogonal Polynomials on one of the search engines - you should find some
interesting material, both mathematical and historical on this general topic.
Also check out Mathematical Induction.
On the arithmetic series, note that
1^k + 2^k + ... + n^k is a polynomial of degree, k+1, where k is an integer.
Now look at the polynomial generated by
a0 + a1(x-1) + a2(x-1)(x-2) + a3(x-1)(x-2)(x-3) + ...
where each "a" is a coefficient.
Substitute, 0 for x, 1 for x, 2 for x, etc. to compute the
appropriate coefficient.
You will be able to develop a unique polynomial to for each one of the
series.
I believe this formula is attributable to Laplace.
There is a more complex method attributable to Lagrange that will
generate coefficients directly for powers of x.
You might want to do a search on these 2 mathematicians as well.
Hope that helps.
(Today, I think the GREEN army is in charge.)
-----
In article <v01540b01b133b361c14a@[206.40.98.24]>,
GARY WARDALL <gwardall@LAKEFIELD.NET> wrote:
>
> Here are two sets of series formulas I developed on the TI-92
> for use on the I-85/86's last week. Some of you find a use for
> these.
>
> Good Luck
>
> Gary Wardall
>
> The FN formulas are for Sum(I^N,I,1,x,1)
> (N=1 is a finite Arithmetic Series.)
>
> F0=x
> F1=x*(x+1)/2
> F2=x*(x+1)*(2*x+1)/6
> F3=x^2*(x+1)^2/4
> F4=x*(x+1)*(2*x+1)*(3*x^2+3*x-1)/30
> F5=x^2*(x+1)^2*(2*x^2+2*x-1)/12
> F6=x/42*(x+1)*(2*x+1)*(3*x^4+6*x^3-3*x+1)
> F7=x^2/24*(x+1)^2*(3*x^4+6*x^3-x^2-4*x+2)
> F8=x/90*(x+1)*(2*x+1)*(5*x^6+15*x^5+5*x^4-15*x^3-x^2+9*x-3)
> F9=x^2*(x+1)^2*(2*x^6+6*x^5+x^4-8*x^3+x^2+6*x-3)/20
>
F10=x/66*(x+1)*(2*x+1)*(3*x^8+12*x^7+8*x^6-18*x^5-10*x^4+24*x^3+2*x^2-15*x+5)
>
> The GN formulas are for Sum(I^N*r^I,I,1,x,1)
> (N=0 is a finite Geometric Series.)
>
> G0=r*r^x/(r-1)-r/(r-1)
> G1=r*r^x*(x+1)/(r-1)-r^2*r^x/(r-1)^2+r/(r-1)^2
>
G2=r*r^x*(x+1)^2/(r-1)-r^2*r^x*(2*x+3)/(r-1)^2+2*r^3*r^x/(r-1)^3-r*(r+1)/(r-1)
^3
> G3=r*r^x*(x+1)^3/(r-1)-r^2*r^x*(3*x^2+9*x+7)/(r-1)^2+6*r^3*r^x*(x+2)/(r-1)^3
> -6*r^4*r^x/(r-1)^4+r*(r^2+4*r+1)/(r-1)^4
> G4=r*r^x*(x+1)^4/(r-1)-r^2*r^x*(4*x^3+18*x^2+28*x+15)/(r-1)^2+2*r^3*r^x*(6*x
> ^2+24*x+25)/(r-1)^3-12*r^4*r^x*(2*x+5)/(r-1)^4+24*r^5*r^x/(r-1)^5-r*(r^3+11*
> r^2+11*r+1)/(r-1)^5
> G5=r*r^x*(x+1)^5/(r-1)-r^2*r^x*(5*x^4+30*x^3+70*x^2+75*x+31)/(r-1)^2+10*r^3*
> r^x*(2*x^3+12*x^2+25*x+18)/(r-1)^3-30*r^4*r^x*(2*x^2+10*x+13)/(r-1)^4+120*r^
> 5*r^x*(x+3)/(r-1)^5-120*r^6*r^x/(r-1)^6+r*(r^4+26*r^3+66*r^2+26*r+1)/(r-1)^6
> G6=r*r^x*(x+1)^6/(r-1)-r^2*r^x*(6*x^5+45*x^4+140*x^3+225*x^2+186*x+63)/(r-1)
> ^2+2*r^3*r^x*(15*x^4+120*x^3+375*x^2+540*x+301)/(r-1)^3-60*r^4*r^x*(2*x^3+15
> *x^2+39*x+35)/(r-1)^4+120*r^5*r^x*(3*x^2+18*x+28)/(r-1)^5-360*r^6*r^x*(2*x+7
> )/(r-1)^6+720*r^7*r^x/(r-1)^7-r*(r^5+57*r^4+302*r^3+302*r^2+57*r+1)/(r-1)^7
> G7=r*r^x*(x+1)^7/(r-1)-r^2*r^x*(7*x^6+63*x^5+245*x^4+525*x^3+651*x^2+441*x+1
> 27)/(r-1)^2+14*r^3*r^x*(3*x^5+30*x^4+125*x^3+270*x^2+301*x+138)/(r-1)^3-42*r
> ^4*r^x*(5*x^4+50*x^3+195*x^2+350*x+243)/(r-1)^4+840*r^5*r^x*(x^3+9*x^2+28*x+
> 30)/(r-1)^5-840*r^6*r^x*(3*x^2+21*x+38)/(r-1)^6+5040*r^7*r^x*(x+4)/(r-1)^7-5
>
040*r^8*r^x/(r-1)^8+r*(r^6+120*r^5+1191*r^4+2416*r^3+1191*r^2+120*r+1)/(r-1)^8
>
> G8=r*r^x*(x+1)^8/(r-1)-r^2*r^x*(8*x^7+84*x^6+392*x^5+1050*x^4+1736*x^3+1764*
> x^2+1016*x+255)/(r-1)^2+2*r^3*r^x*(28*x^6+336*x^5+1750*x^4+5040*x^3+8428*x^2
> +7728*x+3025)/(r-1)^3-84*r^4*r^x*(4*x^5+50*x^4+260*x^3+700*x^2+972*x+555)/(r
> -1)^4+168*r^5*r^x*(10*x^4+120*x^3+560*x^2+1200*x+993)/(r-1)^5-1680*r^6*r^x*(
> 4*x^3+42*x^2+152*x+189)/(r-1)^6+10080*r^7*r^x*(2*x^2+16*x+33)/(r-1)^7-20160*
> r^8*r^x*(2*x+9)/(r-1)^8+40320*r^9*r^x/(r-1)^9-r*(r^7+247*r^6+4293*r^5+15619*
> r^4+15619*r^3+4293*r^2+247*r+1)/(r-1)^9
>
> G9=r*r^x*(x+1)^9/(r-1)-r^2*r^x*(9*x^8+108*x^7+588*x^6+1890*x^5+3906*x^4+5292
> *x^3+4572*x^2+2295*x+511)/(r-1)^2+6*r^3*r^x*(12*x^7+168*x^6+1050*x^5+3780*x^
> 4+8428*x^3+11592*x^2+9075*x+3110)/(r-1)^3-6*r^4*r^x*(84*x^6+1260*x^5+8190*x^
> 4+29400*x^3+61236*x^2+69930*x+34105)/(r-1)^4+504*r^5*r^x*(6*x^5+90*x^4+560*x
> ^3+1800*x^2+2979*x+2025)/(r-1)^5-2520*r^6*r^x*(6*x^4+84*x^3+456*x^2+1134*x+1
> 087)/(r-1)^6+30240*r^7*r^x*(2*x^3+24*x^2+99*x+140)/(r-1)^7-30240*r^8*r^x*(6*
> x^2+54*x+125)/(r-1)^8+362880*r^9*r^x*(x+5)/(r-1)^9-362880*r^10*r^x/(r-1)^10+
> r*(r^8+502*r^7+14608*r^6+88234*r^5+156190*r^4+88234*r^3+14608*r^2+502*r+1)/(
> r-1)^10
>
> G10=r*r^x*(x+1)^10/(r-1)-r^2*r^x*(10*x^9+135*x^8+840*x^7+3150*x^6+7812*x^5+1
> 3230*x^4+15240*x^3+11475*x^2+5110*x+1023)/(r-1)^2+2*r^3*r^x*(45*x^8+720*x^7+
> 5250*x^6+22680*x^5+63210*x^4+115920*x^3+136125*x^2+93300*x+28501)/(r-1)^3-60
> *r^4*r^x*(12*x^7+210*x^6+1638*x^5+7350*x^4+20412*x^3+34965*x^2+34105*x+14575
> )/(r-1)^4+120*r^5*r^x*(42*x^6+756*x^5+5880*x^4+25200*x^3+62559*x^2+85050*x+4
> 9346)/(r-1)^5-2520*r^6*r^x*(12*x^5+210*x^4+1520*x^3+5670*x^2+10870*x+8547)/(
> r-1)^6+15120*r^7*r^x*(10*x^4+160*x^3+990*x^2+2800*x+3047)/(r-1)^7-302400*r^8
> *r^x*(2*x^3+27*x^2+125*x+198)/(r-1)^8+604800*r^9*r^x*(3*x^2+30*x+77)/(r-1)^9
> -1814400*r^10*r^x*(2*x+11)/(r-1)^10+3628800*r^11*r^x/(r-1)^11-r*(r^9+1013*r^
> 8+47840*r^7+455192*r^6+1310354*r^5+1310354*r^4+455192*r^3+47840*r^2+1013*r+1
> )/(r-1)^11
>
> >Does anyone know if there are any programs for the TI-85 that deal with
> >the sums of geometric and arithmetic series', as well as the sums of
> >infinate converging geometric series? Also, are there any good sites
> >that would have large repositories of math-related programs (other than
> >the more well known ones).
> >
> >Thank you,
> >Ratbrain
> >Lord Captain Commander of the Red Army
> >
> >______________________________________________________
> >Get Your Private, Free Email at http://www.hotmail.com
>
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