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Mathematics · unit version 1

Growth estimates and controlled approximation

Prove eventual comparisons, bound harmonic sums and estimate products while tracking the meaning of the error.

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Readiness and preparation

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Does an asymptotic bound describe every small input?

Eventual comparisons require constants and a threshold, while finite guarantees need explicit bounds.

Compare reasoning and find a refresher

No. An O bound holds beyond some threshold with a fixed multiplier. A finite table can suggest it but cannot prove the eventual claim.

Review asymptotic bounds

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How much error does the approximation allow?

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Lesson sequence

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  1. Compare growth and distinguish absolute from relative error

    Lesson details and objectives

    Use eventual inequalities and ratio limits to justify O, Θ, little-o and asymptotic equivalence, including oscillation and cancellation boundaries.

    • State and apply the quantifiers for O, Θ, little-o and asymptotic equivalence on eventually positive sequences.
    • Prove or refute comparison claims using explicit constants, thresholds and subsequences rather than finite plots.
    • Justify safe sum/product rules and diagnose cancellation, logarithm and absolute-error pitfalls.

    10 practice tasks.

  2. Bound harmonic sums and explain logarithmic growth

    Lesson details and objectives

    Compare reciprocal sums with rectangles, control tails and distinguish asymptotic growth from exact thresholds.

    • Prove full-sum and tail bounds using decreasing reciprocal rectangles and verify their indices.
    • Establish divergence with doubling blocks and distinguish sufficient threshold bounds from exact minimum indices.
    • Prove relative logarithmic growth and a positive limiting additive gap with an explicit error bound.

    6 practice tasks.

  3. Estimate products without losing the error

    Lesson details and objectives

    Turn positive products into logarithmic sums, prove factorial area bounds, and use a supplied Stirling remainder with explicit error semantics.

    • Derive two-sided factorial bounds from monotonicity of the natural logarithm.
    • Distinguish equivalence of logarithms, multiplicative equivalence and additive logarithmic error.
    • Use a clearly supplied Stirling remainder theorem to bound relative errors and ratios without forming huge factorials.

    6 practice tasks.