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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The log-free product limit $(1 - 2 / n)^{n} \to \exp (- 2)$

Example

Define a sequence $(a_{n})_{n \in N}$ by $a_{0} : = 0, a_{n} : = {(1 - \frac{2}{ι (n)})}^{n} (n \geq 1) .$ Then $a_{n} ⟶ e^{- 2}$ . The separate value at $n = 0$ avoids division by $ι (0) = 0$ ; the finitely many remaining initial indices with nonpositive base do not affect the limit.

Facts & Assumptions

Given: The sequence $(a_{n})$ defined above.

[L1]

The product-limit theorem holds for every real input once $n > ∣ x ∣$ (For every real $x$ , $(1 + x / n)^{n} \to \exp x$ ).

[L2]

Since $e = \exp (1)$ , the addition and reciprocal formulas and the definition of negative integer powers give $\exp (- 2) = 1 / \exp (2) = 1 / (\exp (1) \exp (1)) = 1 / e^{2} = e^{- 2} > 0$ (The real exponential function and the number $e$ by a power series, The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ , The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , Integer powers $a^{m}$ ).

[L3]

A sequence and any one of its tails have the same limit (Convergence depends only on the tail).

Verification

technique · direct

1.1

For every $n > 2$ , the base is positive and $a_{n} = (1 - 2 / ι (n))^{n}$ , so [L1] applies at $x = - 2$ and gives $a_{n} \to \exp (- 2)$ on that tail.

2.1

By [L3], the whole sequence has the same limit as the tail in step 1.1, and [L2] identifies that limit as $e^{- 2} > 0$ .

step 1.1L2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Analytic Functions (standard reference, not scraped)

The log-free product limit (1−2/n)n→exp⁡(−2)(1−2/n)n→exp(−2)