Hey everyone,
I do not know how to express this integral as a power series:
$\displaystyle \int{\frac{x-arctan(x)}{x^3}$
I cannot figure out how to get this into the right form. I am a bit confused. Any help is appreciated.
Thanks
Hey everyone,
I do not know how to express this integral as a power series:
$\displaystyle \int{\frac{x-arctan(x)}{x^3}$
I cannot figure out how to get this into the right form. I am a bit confused. Any help is appreciated.
Thanks
$\displaystyle \displaystyle \arctan{x} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...$
$\displaystyle \displaystyle x - \arctan{x} = \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} + ...
$
$\displaystyle \displaystyle \frac{x - \arctan{x}}{x^3} = \frac{1}{3} - \frac{x^2}{5} + \frac{x^4}{7} - \frac{x^6}{9} + ...
$
integrate the last series.
why would x be added to each term? ... just subtract the series for arctan(x) from x.
$\displaystyle \displaystyle x - \arctan{x} = x - \left(x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ... \right) = x - x + \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} + ... \, = \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} + ... $