Re: Sum of infinite series - 1/n^2

AI Thread Summary
The discussion centers on computing the sum of the infinite series SUM(X / (Y^X)), where Y is a constant and X is an increasing sequence. Participants clarify that without specific details about the sequence X, it is impossible to determine the sum of the series. The importance of defining both X and Y is emphasized for accurate computation. The conversation highlights the necessity of understanding the nature of the increasing sequence to proceed with the calculation. Overall, the sum cannot be computed without more information about the variables involved.
obelu
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Please I have a similar problem, how can I compute the sum of this infinite series:

SUM(X / (Y^X) ); where X(i)>0
 
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Is this your series?

\sum\frac{X}{Y^X} X(i)>0, what is X, what is Y?
 
Y is a constant, while the X's are increasing from zero to infinity
 
Knowing only that Xn is "some" increasing sequence there is no way to find the sum.
 
I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...

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