English
If x is a square-summable geometric term with |x|<1, then the infinite sum equals (1 - x)^{-1}.
Русский
Если ряд геометрической прогрессии сходится при |x|<1, то сумма равна (1 - x)^{-1}.
LaTeX
$$$\\\\sum_{n=0}^{\\\\infty} x^n = (1 - x)^{-1}, \\\\text{for } |x| < 1$$$
Lean4
/-- `v (1 / (1 + a ^n))` tends to `1` for all `v : AbsoluteValue R S` for fields `R` and `S`,
provided `v a < 1`. -/
theorem tendsto_div_one_add_pow_nhds_one {v : AbsoluteValue R S} {a : R} (ha : v a < 1) :
atTop.Tendsto (fun (n : ℕ) ↦ v (1 / (1 + a ^ n))) (𝓝 1) :=
by
simp_rw [map_div₀ v, v.map_one]
apply one_div_one (G := S) ▸ Tendsto.div tendsto_const_nhds _ one_ne_zero
have h_add := (tendsto_pow_atTop_nhds_zero_of_lt_one (v.nonneg _) ha).const_add 1
have h_sub := (tendsto_pow_atTop_nhds_zero_of_lt_one (v.nonneg _) ha).const_sub 1
exact
tendsto_of_tendsto_of_tendsto_of_le_of_le (by simpa using h_sub) (by simpa using h_add)
(fun n ↦ le_trans (by simp) (v.le_add _ _)) (fun n ↦ le_trans (v.add_le _ _) (by simp))
