# A Series question

• Jul 27th 2014, 07:04 PM
Haytham1111
A Series question
Let f (x) = (x -arctanx) /(x ^3 )for each x ≠ 0 Find the power series with sum equal to f (x) (when | x | ≤ 1, x ≠ 0), and use this power series to calculate the limit limx → 0f (x).
• Jul 27th 2014, 08:31 PM
SlipEternal
Re: A Series question
What have you tried? Do you know the power series for $\displaystyle \arctan x$? Have you tried just plugging that in and simplifying?
• Jul 27th 2014, 08:55 PM
Haytham1111
Re: A Series question
Yes i did, i got series (-1)^(n+1) *x^(2n-2) (series from n=1 to infinity)
But it was not the correct answer

The correctly answer is ∑k=0∞((−1)^k)/(2k+3)*x^(2k)

1/3

I do not understand how he get this answer. Have you any suggestion?
• Jul 27th 2014, 09:25 PM
SlipEternal
Re: A Series question
$\displaystyle x-\arctan x = x - \sum_{n=0}^\infty (-1)^n \dfrac{x^{2n+1}}{2n+1}$

The first term of the $\displaystyle \arctan x$ series is x. So, when you perform the subtraction, you get:

$\displaystyle x-\arctan x = \sum_{n=0}^\infty (-1)^n \dfrac{x^{2n+3}}{2n+3}$

Then, dividing by $\displaystyle x^3$ gives the series

$\displaystyle \dfrac{x-\arctan x}{x^3} = \sum_{n=0}^\infty (-1)^n \dfrac{x^{2n}}{2n+3}$
• Jul 28th 2014, 03:47 AM
Haytham1111
Re: A Series question
Thank you. It is the same i got, but i should reindeks the series to start from n=0 .

I am leserinnlegg Latex and i use texmaker. How i can write the dokument in latex in this forum. Can you help?
• Jul 28th 2014, 05:47 AM
SlipEternal
Re: A Series question
Quote:

Originally Posted by Haytham1111
Thank you. It is the same i got, but i should reindeks the series to start from n=0 .

I am leserinnlegg Latex and i use texmaker. How i can write the dokument in latex in this forum. Can you help?

Click Reply With Quote on any post with LaTeX to see how it is formatted on this forum.