MacLaurin k/ k^2+1

Question

The MacLaurin series for $\frac{1}{1+k}$ $=\sum_{n=0}^\infty (-1)^n k^n$

use the series above to find the MacLaurin series for $\frac{k}{1+k^2}$

I found the series is $=\sum_{n=0}^\infty (-1)^n k^{2n+1}$ but I'm not sure how can we use the first series to find out the series for $\frac{k}{1+k^2}$

Answer 1

$$\frac{1}{1+k}=\sum_{n=0}^\infty (-1)^n k^n$$ let $k\to k^2$ $$\frac{1}{1+k^2}=\sum_{n=0}^\infty (-1)^n k^{2n}$$ then multiply by $k$

Answer 2

in the first expansion, you can replace $k$ by $k^2$ since

$k^2$ tends to $0$ when $k$ goes to 0.

then you multiply by $k$.

your result is correct.

YOU CAN NOT REPLACE k by something which does not go to 0 when k goes to 0.!!