summable_log_one_add_of_summable — Mathlib

English

If a real-valued function f on an index set is summable and each f_i ≠ 0, then log(f_i) is summable and the corresponding product exists.

Русский

Если сумма log f_i сходится и все f_i ≠ 0, то ∑ log f_i сходится и произведение существует.

LaTeX

$$$\text{HasProd}(f) = \text{exists}$ and the sum of logs converges; hence the product exists.$$

Lean4

theorem summable_log_one_add_of_summable {f : ι → ℂ} (hf : Summable f) : Summable (fun i ↦ log (1 + f i)) :=
  by
  apply (hf.norm.mul_left (3 / 2)).of_norm_bounded_eventually
  filter_upwards [hf.norm.tendsto_cofinite_zero.eventually_le_const one_half_pos] with i hi using
    norm_log_one_add_half_le_self hi

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