### Nuprl Lemma : cosine-poly-approx-1

`∀[x:{x:ℝ| (r0 ≤ x) ∧ (x ≤ r1)} ]. ∀[k:ℕ].`
`  (|cosine(x) - Σ{-1^i * (x^2 * i)/(2 * i)! | 0≤i≤k}| ≤ (x^(2 * k) + 2/r(((2 * k) + 2)!)))`

Proof

Definitions occuring in Statement :  cosine: `cosine(x)` rsum: `Σ{x[k] | n≤k≤m}` rdiv: `(x/y)` rleq: `x ≤ y` rabs: `|x|` rnexp: `x^k1` int-rdiv: `(a)/k1` int-rmul: `k1 * a` rsub: `x - y` int-to-real: `r(n)` real: `ℝ` fastexp: `i^n` fact: `(n)!` nat: `ℕ` uall: `∀[x:A]. B[x]` and: `P ∧ Q` set: `{x:A| B[x]} ` multiply: `n * m` add: `n + m` minus: `-n` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than: `a < b` squash: `↓T` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` subtype_rel: `A ⊆r B` so_apply: `x[s]` rneq: `x ≠ y` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` nat_plus: `ℕ+` uiff: `uiff(P;Q)` series-sum: `Σn.x[n] = a` converges-to: `lim n→∞.x[n] = y` sq_exists: `∃x:A [B[x]]` less_than': `less_than'(a;b)` rless: `x < y` sq_stable: `SqStable(P)` rleq: `x ≤ y` rnonneg: `rnonneg(x)` real: `ℝ` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` top: `Top` rdiv: `(x/y)` req_int_terms: `t1 ≡ t2` cand: `A c∧ B` subtract: `n - m` true: `True` int_upper: `{i...}` sq_type: `SQType(T)` nequal: `a ≠ b ∈ T ` int_nzero: `ℤ-o`

Latex:
\mforall{}[x:\{x:\mBbbR{}|  (r0  \mleq{}  x)  \mwedge{}  (x  \mleq{}  r1)\}  ].  \mforall{}[k:\mBbbN{}].
(|cosine(x)  -  \mSigma{}\{-1\^{}i  *  (x\^{}2  *  i)/(2  *  i)!  |  0\mleq{}i\mleq{}k\}|  \mleq{}  (x\^{}(2  *  k)  +  2/r(((2  *  k)  +  2)!)))

Date html generated: 2020_05_20-AM-11_26_22
Last ObjectModification: 2019_12_14-PM-04_51_41

Theory : reals

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