Show that $\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1$ 
Show that $$\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1 .$$

I know that $\arctan 1 = \frac{\pi}{4}$ and that the sequence being subtracted is a partial sum of its Taylor series. I believe you use the alternating series test to explain, but all I get from it is that the series will converge on $[-1,1]$.

you can actually get a stronger result such as $$\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 1/11 $$
because the error in a convergent alternating series is less than the absolute value of the  first missing term which in this case is $1/11.$
The alternating series is simply the Taylor series of $\tan ^{-1} x $ evaluated at $x=1$ and it converges by the alternating series test. 
