Nuprl Lemma : fun-converges-to-sine
lim n→∞.Σ{-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! | 0≤i≤n} = λx.sine(x) for x ∈ (-∞, ∞)
Proof
Definitions occuring in Statement :
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
riiint: (-∞, ∞),
sine: sine(x),
rsum: Σ{x[k] | n≤k≤m},
rnexp: x^k1,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
fastexp: i^n,
fact: (n)!,
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
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,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
nat_plus: ℕ+,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
le: A ≤ B,
int_upper: {i...},
rless: x < y,
sq_exists: ∃x:A [B[x]],
sq_stable: SqStable(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
series-sum: Σn.x[n] = a,
real: ℝ,
subtract: n - m,
top: Top,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
sq_type: SQType(T),
rge: x ≥ y
Latex:
lim n\mrightarrow{}\minfty{}.\mSigma{}\{-1\^{}i * (x\^{}(2 * i) + 1)/((2 * i) + 1)! | 0\mleq{}i\mleq{}n\} = \mlambda{}x.sine(x) for x \mmember{} (-\minfty{}, \minfty{})
Date html generated:
2020_05_20-PM-01_09_35
Last ObjectModification:
2020_01_03-AM-00_17_07
Theory : reals
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