Math Is Fun Forum

mikau

Member

Registered: 2005-08-22

Posts: 1,504

power series

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)

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A logarithm is just a misspelled algorithm.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: power series

nevermind I found out what it was. Appearently, when differentiating a power series, the index has to shift up. Hmm...wonder why...

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A logarithm is just a misspelled algorithm.

HallsofIvy

Guest

Re: power series

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

.