Re: simple series

On Wed, 06 Aug 2008 17:07:06 EDT, amy666 <tommy1729@xxxxxxxxxxx>
wrote:

fabri wrote :

Hi,

I have the series

sum_{n from 1 to infty;n odd}[(n+1)x^n].

Using software, I see that when |x|<1 the series
converges to 2x/(1-x^2)^2. Anyone so kind to explain
me how you get the sum analytically?

thank you

taylor taylor taylor taylor taylor

did i mention taylor ?

You're given the sum. You're _not_ given that
it equals 2x/(1-x^2)^2. How do you deduce
from Taylor's theorem that the sum equals
2x/(1-x^2)^2, without cheating?

The other replies in the thread give much better
methods, because they explain how a person could
_find_ the expression 2x/(1-x^2)^2.

Second question:

Suppose you're given the expression 2x/(1-x^2)^2.
How do you use Taylor's theorem to show that
it equals that sum?

Don't bother showing how to calculate the derivatives.
Let's take as given that that sum is in fact the
Taylor series for the function. Now how do you
show that the sum is actually _equal_ to that
function?

This one is not too hard, but you can't do it because
you don't know the theory involved. Note:

(i) The Taylor series for an infinitely differentiable
function need not converge.

(ii) If the Taylor series for f _does_ converge
it can converge to something other than f.

So it's simply _wrong_ to just assume that a
given function is equal to its Taylor series.
Hence the argument you had in mind when
you said "taylor taylor taylor" is far from
a complete proof. (If a person actually knows
some math filling in this proof is not all that
hard, like I said. But the other posts in the
thread are much simpler than a _correct_
solution by Taylor's theorem.)

Oops. I said you don't know the theory required
to give a correct proof by Taylor's theorem.
How do I know whether you know that or not?
So prove me wrong - show us that proof.
David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

Relevant Pages

  • Re: Series problem
    ... which makes your sum ... I use Taylor axpansion (or maybe it's ... > Sorry if I am asking simple questions, but I have an exam tomorrow and I ...
    (comp.sys.hp48)
  • Re: simple series
    ... taylor taylor taylor taylor taylor ... the OP gave the statement that it equals 2x/^2. ... well it matches its taylor series, so all we need now is the radius of convergeance. ... "Understanding Godel isn't about following his formal ...
    (sci.math)
  • Re: Taylor Series (Complex)
    ... so each one is the sum of its Taylor series). ... David C. Ullrich ...
    (sci.math)
  • Re: solving an equation
    ... >Don Taylor wrote: ... Thank you for pointing out my mistake, ... I glanced at this and then did a SUM, ...
    (sci.math)
  • Re: simple series
    ... taylor taylor taylor taylor taylor ... bla bla bla, i just did it. ... and although you David claim the others threaded ... That would make a mockery of everything Godel ...
    (sci.math)