# Chapter 25: How to Solve Estimation Problems

> When is your approximation actually accurate? Remainder bounds, alternating-series estimates, and five fully-worked estimation problems.

**Canonical:** https://calculus.kitzo.work/chapter/taylor-estimation  
**Level:** Advanced  
**Estimated time:** 45 minutes  
**Exercises:** 8 (auto-marked; answers stay server-side)

## Topics
- Taylor error term
- Bounding the error in practice
- Estimating eˣ and trig values
- Alternating series estimation
- Estimating integrals by series
- General tactics

**Why this matters**

“Approximately” is not an answer until you say how approximate. This chapter turns Taylor’s remainder into a rigorous error bound — and shows you the fast trick (Alternating Series Estimation) when it applies.

## 1 · The error term, properly

After Pₙ(x), the error is f(x) − Pₙ(x) = f⁽ⁿ⁺¹⁾(ξ)·(x−a)ⁿ⁺¹/(n+1)!. If we can bound |f⁽ⁿ⁺¹⁾| ≤ M on our interval, then |error| ≤ M·|x−a|ⁿ⁺¹/(n+1)! — a hard guarantee.

## 2 · First example — estimating e^{0.1}

Use P₂(0.1) = 1.10517. Remainder ≤ e^{0.1}(0.1)³/6 ≈ 1.84×10⁻⁴. So e^{0.1} = 1.10517 ± 0.00018. Three correct decimals, proven.

## 3 · Second example — a trig estimate

Estimate sin(0.3) using P₃(x) = x − x³/6: P₃(0.3) = 0.3 − 0.0045 = 0.2955. The next term is +x⁵/120 = 0.00002, so error ≤ 2.1×10⁻⁵. Answer: 0.2955 ± 0.00002.

## 4 · Third example — logarithms

Estimate ln(1.1) ≈ 0.1 − 0.005 + 0.00033 ≈ 0.09533. Remainder over the geometric-like alternating tail is below the first omitted term 0.1⁴/4 = 2.5×10⁻⁵. So 0.09533 ± 0.00003.

## 5 · Fourth example — an integral by series

∫₀^{0.5} e^{−x²}dx. Expand e^{−x²} = 1 − x² + x⁴/2 − ⋯, integrate term-by-term: 0.5 − 0.125/3 + 0.03125/10 − ⋯ ≈ 0.4615. Bounding the tail by the first omitted term keeps error < 0.0001.

**Worked example 1**

**Estimate ∫₀^{0.5} e^{−x²}dx by integrating the series.** Expand e^{−x²} = 1 − x² + x⁴/2 − ⋯, integrate term-by-term: 0.5 − 0.125/3 + 0.03125/10 − ⋯ ≈ 0.4615. Bounding the tail by the first omitted term keeps the error under 0.0001 — a concrete instance of an estimation problem.

## 6 · The alternating-series shortcut

If a series alternates with terms decreasing to 0, the error is *less than the first term you omitted*. No M-hunting needed — often the fastest rigorous bound in the course.

**Watch out**

Declaring “the series converges” is not an error bound. You must exhibit M and the factorial denominator, or cite the alternating test with decreasing terms. A value without a bound is a rumor, not a result.

**Chapter formula sheet**

Lagrange: |error| ≤ M·|x−a|ⁿ⁺¹/(n+1)!  (M = max |f⁽ⁿ⁺¹⁾| on interval)

Alternating: |f(x) − Pₙ(x)| ≤ |first omitted term|

Pick M generously (e^{|x|} ≤ e^{0.5} on |x|≤0.5, |sin| ≤ 1, |cos| ≤ 1)

Rule of thumb: 3–4 terms usually buy 4+ decimals near 0

## Common mistakes

- Waving “the series converges” without bounding M — convergence ≠ enough accuracy.

- Forgetting the factorial in the remainder — off by a factor of (n+1) is a failed proof.

- Using an alternating bound when the series does not alternate — the shortcut is licensed only there.

- Confusing “error” sign: remainder can be positive or negative; the bound is on its size.

## Practice exercises

1. The Lagrange error bound needs:

- A. M = max |f⁽ⁿ⁺¹⁾| on the interval
- B. the radius of convergence only
- C. the alternating series test
- D. the value of f⁽ⁿ⁺¹⁾ at a

2. Using sin x ≈ x, bound |sin x − x| at x = 0.1 (next term x³/6). Give an upper bound.

3. e^{0.1} via P₂ is 1.10517 with remainder ≤ ___. Give a safe upper bound (decimal).

4. Alternating Series Estimation: error ≤

- A. the first omitted term
- B. twice the first omitted term
- C. the last used term
- D. the sum of the tails

5. Using cos x ≈ 1 − x²/2, bound the error at x = 0.4 (next term x⁴/24).

6. To bound e^{−x²} on [0, 0.5] for an integral estimate, a good M is:

- A. 1 (since e^{−x²} ≤ 1)
- B. 1000
- C. 0.5
- D. you don’t need M with alternating series

7. ln(1.1) ≈ 0.1 − 0.005 + 0.00033. The next term’s size is:

8. The fifth classic estimation problem is:

- A. Estimating an integral by term-by-term integration with an error bound
- B. Proving l’Hôpital’s rule
- C. Finding arc length exactly
- D. Solving a differential equation

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