Estimating the Value of a Series — Question 1

*Question 1 Let $\displaystyle S=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}n$ and $S_N=\sum_{n=1}^{N}(-1)^{n-1}/n$.

1. State the Alternating Series Estimation Theorem for $R_N=S-S_N$.

2. Find the least final index $N$ for which the next-term bound $1/(N+1)$ is strictly below $10^{-3}$.

*Question 1 – Solution

The magnitudes $b_n=1/n$ decrease to $0$, so the alternating series converges and $$|R_N|\le b_{N+1}=\frac1{N+1}.$$

$$\frac1{N+1}<10^{-3}\iff N+1>1000\iff N>999.$$ The least integer is $\boxed{N=1000}$.

For $N=999$, the displayed non-strict bound gives $|R_{999}|\le1/1000=10^{-3}$, so this bound is not strictly below the tolerance. (The actual remainder is strictly smaller than the next term.) For $N=1000$, $|R_{1000}|\le1/1001<10^{-3}$.

Use the first $1000$ terms, ending at index $1000$.