### Nuprl Lemma : triangular-reciprocal-series-sum

`Σn.(r1/r(t(n + 1))) = r(2)`

This theorem is one of freek's list of 100 theorems

Proof

Definitions occuring in Statement :  series-sum: `Σn.x[n] = a` rdiv: `(x/y)` int-to-real: `r(n)` triangular-num: `t(n)` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_apply: `x[s]` nat: `ℕ` uimplies: `b supposing a` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` ge: `i ≥ j ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` converges-to: `lim n→∞.x[n] = y` sq_exists: `∃x:A [B[x]]` nat_plus: `ℕ+` le: `A ≤ B` uiff: `uiff(P;Q)` subtract: `n - m` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` true: `True` rdiv: `(x/y)` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` rev_uimplies: `rev_uimplies(P;Q)` triangular-num: `t(n)` divide: `n ÷ m` series-sum: `Σn.x[n] = a` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` pointwise-req: `x[k] = y[k] for k ∈ [n,m]` nequal: `a ≠ b ∈ T ` int_nzero: `ℤ-o` rsub: `x - y` rat_term_to_real: `rat_term_to_real(f;t)` rtermAdd: `left "+" right` rat_term_ind: rat_term_ind rtermConstant: `"const"` rtermDivide: `num "/" denom` rtermVar: `rtermVar(var)` pi1: `fst(t)` rtermMinus: `rtermMinus(num)` pi2: `snd(t)`
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity dependent_functionElimination thin sqequalRule lambdaEquality isectElimination natural_numberEquality hypothesis addEquality setElimination rename because_Cache independent_isectElimination inrFormation productElimination independent_functionElimination hypothesisEquality unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll lambdaFormation dependent_set_memberFormation dependent_set_memberEquality multiplyEquality functionEquality applyEquality minusEquality lambdaFormation_alt dependent_set_memberEquality_alt approximateComputation dependent_pairFormation_alt lambdaEquality_alt isect_memberEquality_alt universeIsType equalityTransitivity equalitySymmetry equalityIstype baseApply closedConclusion baseClosed sqequalBase inrFormation_alt imageElimination inhabitedIsType imageMemberEquality universeEquality instantiate

Latex:
\mSigma{}n.(r1/r(t(n  +  1)))  =  r(2)

Date html generated: 2019_10_29-AM-10_25_26
Last ObjectModification: 2019_04_02-AM-10_01_11

Theory : reals

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