33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.


π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

Verified step by step guidance

1. Recognize that the given series is an alternating series of the form \(\sum_{k=0}^\infty (-1)^k \frac{1}{2k+1}\), which converges to \(\frac{\pi}{4}\).
2. Recall the Alternating Series Remainder Theorem, which states that the magnitude of the remainder \(R_n\) after summing \(n\) terms is less than or equal to the absolute value of the first omitted term: \(|R_n| \leq \left| a_{n+1} \right|\).
3. Identify the \((n+1)\)-th term of the series: \(a_{n+1} = \frac{1}{2(n+1) + 1} = \frac{1}{2n + 3}\).
4. Set up the inequality to ensure the remainder is less than \$10^{-4}\(: \)\frac{1}{2n + 3} < 10^{-4}$.
5. Solve this inequality for \(n\) to find the minimum number of terms needed: multiply both sides by \$2n + 3\(, then isolate \)n\( and solve for the smallest integer \)n$ satisfying the inequality.

Key Concepts

Alternating Series and Convergence

An alternating series is a series whose terms alternate in sign, typically of the form (−1)^k * a_k with a_k > 0. Such series can converge if the terms decrease in magnitude to zero. Understanding this helps determine when the infinite sum approaches a finite limit.

Alternating Series Remainder (Error) Estimation

The remainder after summing n terms of a convergent alternating series is less than or equal to the magnitude of the first omitted term. This property allows us to estimate how many terms are needed to ensure the error is below a desired threshold.

Partial Sums and Error Bounds

A partial sum is the sum of the first n terms of a series. For alternating series, the error bound is given by the absolute value of the (n+1)-th term. Using this, one can find the minimum n such that the remainder is less than a specified small number, like 10⁻⁴.