Estimating the Value of a Series — Question 1

Question 1 – Solution

Step 1: Verify the theorem applies.

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

Step 2: Solve the strict inequality.

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

Step 3: Check minimality.

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 $N=1000$, $|R_{1000}|\le1/1001<10^{-3}$.

Conclusion.

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