How many terms of this series would one need to add to get an approximation of $\pi$ with error less than $10^{-4}$?
$$ 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots $$
So far, I wrote the series as $$\sum\limits_{n=0}^\infty (-1)^n \frac{4}{2n+1}$$
and will use $|S-S_n|< a_n + 1$ , but am not sure how to go about doing so from here.
On
The alternating series theorem says that the error is less than the absolute value of the first neglected term, so you just need to find the first term less than $10^{-4}$. You don't know that this term is the first one for which the error is less than $10^{-4}$, but you know it works.
Let $\displaystyle S_n = 4\sum_{k=0}^n \frac{(-1)^k}{2k+1}$.
Then
$$ |S_{10001}-S_{10000}| = |a_{10001}| = \left|\frac{-4}{20003}\right| < 0.0002.\tag 1 $$ We know that $S_{10000}>\pi>S_{10001}$ but $\pi$ is somewhat closer to the latter term than to the former, and $S_{10002}>\pi>S_{10001}$ and $\pi$ is somewhat closer to the former term than to the latter. Make this an exercise: $|S_n-\pi|$ decreases as $n$ grows; that justifies my former two statements.
Thus $\pi$ is half-way between $S_{10001}$ and some number that is between $S_{10000}$ and $S_{10002}$. So $|\pi-S_{10002}|$ is less than half the difference between $S_{10000}$ and $S_{10001}$, and that difference is given in $(1)$.
The error is about half the size of the next term.