1. ## 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).

2. ## Re: A Series question

What have you tried? Do you know the power series for $\arctan x$? Have you tried just plugging that in and simplifying?

3. ## 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

1/3

I do not understand how he get this answer. Have you any suggestion?

4. ## Re: A Series question

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

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

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

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

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

5. ## 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?

6. ## Re: A Series question

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?
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