changeOriginSeries_finite_of_finite

English

The sum of changeOrigin series equals partial sums and recovers the original sum in finite cases: (p.changeOrigin k).sum = (p.changeOrigin k).partialSum (n - k).

Русский

Сумма ряда изменения соответствует частичной сумме и восстанавливает исходную сумму в конечном случае.

LaTeX

$$$(p.changeOrigin k).sum = (p.changeOrigin k).partialSum (n - k)$$

Lean4

/-- If `p` is a finite formal multilinear series, then so is `p.changeOriginSeries k` for every
`k` in `ℕ`. More precisely, if `p m = 0` for `n ≤ m`, then `p.changeOriginSeries k m = 0` for
`n ≤ k + m`. -/
theorem changeOriginSeries_finite_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ) : ∀ {m : ℕ}, n ≤ k + m → p.changeOriginSeries k m = 0 :=
  by
  intro m hm
  rw [changeOriginSeries]
  exact Finset.sum_eq_zero (fun _ _ => p.changeOriginSeriesTerm_bound hn _ _ _ hm)

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