od peterblack » 29. 6. 2008 23:08
ad priklady od drahose:
Sum[ (Sin[x]Sin[nx]) / Sqrt[n+x] ] na intervalu <0,2Pi)
u tohodle by vam melo pomoct:
http://forum.matweb.cz/viewtopic.php?id=1976
http://kam.mff.cuni.cz/%7Eklazar/analyzaII.pdf (strana 35)
btw nasel jsem na netu reseni toho souctu sumy 1/(n^2(n+1)^2) :
a(n) being equal to 1/(n^2(n+1)^2) spits into partial fractions:
A/n+B/n^2+C/(n+1)+D/(n+1)^2, where
A,B,C,D are yet undetermined coefficients which,when found,
are equal to -2, -2, 1, 1 respectively.
Thus a(n)={-2/n+2/(n+1)}+1/n^2+1/(n+1)^2,
and the Euler's series summation(1/n^2,n=1..infinity)=Pi^2/6, then
Then summation(a(n),n=1..infinity)=
={-2/1+2/2-2/2+2/3-2/4+...}+
+Pi^2/6+[Pi^2/6-1]
=-2-1+2*Pi^2/6
=-3+Pi^2/3.
That is, the two series in braces cancel each over out resulting in -2. On the other hand, the series
summation(1/(n+1)^2)=1/2^2+1/3^2+..
=(-1)+(1+1/2^2+1/3^2+..)
=(-1)+Pi^2/6.
http://answers.yahoo.com/question/index ... 551AAnnoTq
a tady je popsanej problem suma 1/n^2=Pi^2/6 (Euler's summation formula):
http://en.wikipedia.org/wiki/Basel_prob ... he_problem
http://www.math.nmsu.edu/~davidp/euler2k2.pdf
btw na pochopeni prikladu z fourierovek je nejlepsi tohle...mnoho obecnych praktickych vzorcu:
http://mathonline.fme.vutbr.cz/Fouriero ... fault.aspx
a tady na eulerovy substituce:
http://www.mojeskola.cz/Vyuka/Php/Learn ... okem10.php
ad priklady od drahose:
Sum[ (Sin[x]Sin[nx]) / Sqrt[n+x] ] na intervalu <0,2Pi)
u tohodle by vam melo pomoct:
http://forum.matweb.cz/viewtopic.php?id=1976
http://kam.mff.cuni.cz/%7Eklazar/analyzaII.pdf (strana 35)
btw nasel jsem na netu reseni toho souctu sumy 1/(n^2(n+1)^2) :[quote]
a(n) being equal to 1/(n^2(n+1)^2) spits into partial fractions:
A/n+B/n^2+C/(n+1)+D/(n+1)^2, where
A,B,C,D are yet undetermined coefficients which,when found,
are equal to -2, -2, 1, 1 respectively.
Thus a(n)={-2/n+2/(n+1)}+1/n^2+1/(n+1)^2,
and the Euler's series summation(1/n^2,n=1..infinity)=Pi^2/6, then
Then summation(a(n),n=1..infinity)=
={-2/1+2/2-2/2+2/3-2/4+...}+
+Pi^2/6+[Pi^2/6-1]
=-2-1+2*Pi^2/6
=-3+Pi^2/3.
That is, the two series in braces cancel each over out resulting in -2. On the other hand, the series
summation(1/(n+1)^2)=1/2^2+1/3^2+..
=(-1)+(1+1/2^2+1/3^2+..)
=(-1)+Pi^2/6.
[/quote]
http://answers.yahoo.com/question/index?qid=20080619044551AAnnoTq
a tady je popsanej problem suma 1/n^2=Pi^2/6 (Euler's summation formula):
http://en.wikipedia.org/wiki/Basel_problem#Euler_attacks_the_problem
http://www.math.nmsu.edu/~davidp/euler2k2.pdf
btw na pochopeni prikladu z fourierovek je nejlepsi tohle...mnoho obecnych praktickych vzorcu:
http://mathonline.fme.vutbr.cz/Fourierovy-rady/sc-73-sr-1-a-60/default.aspx
a tady na eulerovy substituce:
http://www.mojeskola.cz/Vyuka/Php/Learning/Derivace/matika_krokem10.php