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ns36
 2 years ago
Help with sequences and series:
Show that the function f(x) = sum from 0 to infinity ((1)^n * x^(2n))/((2n)!) is a solution of the differential equation f''(x) + f(x) = 0.
ns36
 2 years ago
Help with sequences and series: Show that the function f(x) = sum from 0 to infinity ((1)^n * x^(2n))/((2n)!) is a solution of the differential equation f''(x) + f(x) = 0.

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ns36
 2 years ago
Best ResponseYou've already chosen the best response.1I'll try to make to write f(x) in the equation editor....one sec.

ns36
 2 years ago
Best ResponseYou've already chosen the best response.1\[\sum_{0}^{\infty} ((1)^{n} x^{2n})/((2n)!)\]

ns36
 2 years ago
Best ResponseYou've already chosen the best response.1That's f(x), and I need to show that it's a solution to the differential equation f''(x) + f(x) = 0.

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1take the second derivative of the sumation and add it to the sumation itself to see if it goes to 0

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1the summation is jsut a polynomial

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1in teh eq editor, if you type in \frac{}{} you can fill in the {}s with your top and bottom arguments

ns36
 2 years ago
Best ResponseYou've already chosen the best response.1Oh I see, f(x) is just cos(x)! Thanks a lot!

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1\[\sum_{0}^{\infty} \frac{(1)^{n} x^{2n}}{(2n)!}\] \[D_x[\sum ]=\sum_{1}^{\infty} \frac{(1)^{n} 2n\ x^{(2n1)}}{(2n)!}\] \[D_{xx}[\sum ]=\sum_{2}^{\infty} \frac{(1)^{n} 2n(2n1)\ x^{(2n2)}}{(2n)!}\]

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1i think i need more practice on my summation derivatives :) but your way is sufficient i believe
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