Convergent series. Is my logic correct?

[emilkh]

Show $$\sum_1^\infty\frac{x^n}{1+x^n}$$ converges when x is in [0,1)
$$\sum_1^\infty\frac{x^n}{1+x^n} = \sum_1^\infty\frac{1}{1+x^n} * x^n <= \sum_1^\infty\frac{1}{1} * x^n = \sum_1^\infty x^n$$

The last sum is g-series, converges since r = x < 1

 

[HallsofIvy]

Yes, that is correct. In fact, you can say more: the series converges for x in (-1, 1).

 

[mathwonk]

ratio test?