Nuprl Lemma : partial-geometric-series
∀n:ℕ. ∀c:ℝ. (c ≠ r1 ⇒ (Σ{c^i | 0≤i≤n} = (r1 - c^n + 1/r1 - c)))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rneq: x ≠ y,
rnexp: x^k1,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
nat_plus: ℕ+,
subtract: n - m,
sq_type: SQType(T),
guard: {T},
less_than': less_than'(a;b),
lt_int: i <z j,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
subtype_rel: A ⊆r B,
rdiv: (x/y)
Latex:
\mforall{}n:\mBbbN{}. \mforall{}c:\mBbbR{}. (c \mneq{} r1 {}\mRightarrow{} (\mSigma{}\{c\^{}i | 0\mleq{}i\mleq{}n\} = (r1 - c\^{}n + 1/r1 - c)))
Date html generated:
2020_05_20-AM-11_21_38
Last ObjectModification:
2020_01_02-PM-02_11_05
Theory : reals
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