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use differentiation to find a power series representation for
f(x) = 1/(1+x)^2
where |x| < 1.
if g(x) = -1/(1 + x) then g'(x) = f(x).
but g(x) = -1/(1 + x) = - 1/(1 - (-x)) = ∑ (-1)^(n+1)x^n from n = 0 to infinity
so we can differentiatate the series term by term to find f(x). So we get
∑ n(-1)^(n+1)x^(n-1) from n = 0 to infinity.
however, my book has it written as ∑ (n+1)(-1)^(n+1)x^n from n = 0 to infinity.
Only differences there is the indices are shifted up one.
Any idea why?
Last edited by mikau (2007-05-05 18:16:50)
A logarithm is just a misspelled algorithm.
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nevermind I found out what it was. Appearently, when differentiating a power series, the index has to shift up. Hmm...wonder why...
A logarithm is just a misspelled algorithm.
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nevermind I found out what it was. Appearently, when differentiating a power series, the index has to shift up. Hmm...wonder why...
Because you are "indexing" by the power of x and the decreases when you differentiate. For example, the derivative of
is and each term is of the form .Pages: 1