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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | geolim 14601* | The partial sums in the infinite series ... converge to . (Contributed by NM, 15-May-2006.) |
Theorem | geolim2 14602* | The partial sums in the geometric series ... converge to . (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Theorem | georeclim 14603* | The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Theorem | geo2sum 14604* | The value of the finite geometric series ... , multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Theorem | geo2sum2 14605* | The value of the finite geometric series ... . (Contributed by Mario Carneiro, 7-Sep-2016.) |
..^ | ||
Theorem | geo2lim 14606* | The value of the infinite geometric series ... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.) |
Theorem | geomulcvg 14607* | The geometric series converges even if it is multiplied by to result in the larger series . (Contributed by Mario Carneiro, 27-Mar-2015.) |
Theorem | geoisum 14608* | The infinite sum of ... is . (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Theorem | geoisumr 14609* | The infinite sum of reciprocals ... is . (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Theorem | geoisum1 14610* | The infinite sum of ... is . (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Theorem | geoisum1c 14611* | The infinite sum of ... is . (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Theorem | 0.999... 14612 | The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. , is exactly equal to 1, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.) |
; | ||
Theorem | 0.999...OLD 14613 | Obsolete version of 0.999... 14612 as of 8-Sep-2021. (Contributed by NM, 2-Nov-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | geoihalfsum 14614 | Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 14610. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 14612 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.) |
Theorem | cvgrat 14615* | Ratio test for convergence of a complex infinite series. If the ratio of the absolute values of successive terms in an infinite sequence is less than 1 for all terms beyond some index , then the infinite sum of the terms of converges to a complex number. Equivalent to first part of Exercise 4 of [Gleason] p. 182. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 27-Apr-2014.) |
Theorem | mertenslem1 14616* | Lemma for mertens 14618. (Contributed by Mario Carneiro, 29-Apr-2014.) |
Theorem | mertenslem2 14617* | Lemma for mertens 14618. (Contributed by Mario Carneiro, 28-Apr-2014.) |
Theorem | mertens 14618* | Mertens' theorem. If is an absolutely convergent series and is convergent, then (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.) |
Theorem | prodf 14619* | An infinite product of complex terms is a function from an upper set of integers to . (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | clim2prod 14620* | The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.) |
Theorem | clim2div 14621* | The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.) |
Theorem | prodfmul 14622* | The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) |
Theorem | prodf1 14623 | The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.) |
Theorem | prodf1f 14624 | A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.) |
Theorem | prodfclim1 14625 | The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.) |
Theorem | prodfn0 14626* | No term of a nonzero infinite product is zero. (Contributed by Scott Fenton, 14-Jan-2018.) |
Theorem | prodfrec 14627* | The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.) |
Theorem | prodfdiv 14628* | The quotient of two infinite products. (Contributed by Scott Fenton, 15-Jan-2018.) |
Theorem | ntrivcvg 14629* | A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.) |
Theorem | ntrivcvgn0 14630* | A product that converges to a nonzero value converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.) |
Theorem | ntrivcvgfvn0 14631* | Any value of a product sequence that converges to a nonzero value is itself nonzero. (Contributed by Scott Fenton, 20-Dec-2017.) |
Theorem | ntrivcvgtail 14632* | A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.) |
Theorem | ntrivcvgmullem 14633* | Lemma for ntrivcvgmul 14634. (Contributed by Scott Fenton, 19-Dec-2017.) |
Theorem | ntrivcvgmul 14634* | The product of two non-trivially converging products converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.) |
Syntax | cprod 14635 | Extend class notation to include complex products. |
Definition | df-prod 14636* | Define the product of a series with an index set of integers . This definition takes most of the aspects of df-sum 14417 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodex 14637 | A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq1f 14638 | Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.) |
Theorem | prodeq1 14639* | Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.) |
Theorem | nfcprod1 14640* | Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | nfcprod 14641* | Bound-variable hypothesis builder for product: if is (effectively) not free in and , it is not free in . (Contributed by Scott Fenton, 1-Dec-2017.) |
Theorem | prodeq2w 14642* | Equality theorem for product, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq2ii 14643* | Equality theorem for product, with the class expressions and guarded by to be always sets. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq2 14644* | Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | cbvprod 14645* | Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | cbvprodv 14646* | Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | cbvprodi 14647* | Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq1i 14648* | Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq2i 14649* | Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq12i 14650* | Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq1d 14651* | Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq2d 14652* | Equality deduction for product. Note that unlike prodeq2dv 14653, may occur in . (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq2dv 14653* | Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq2sdv 14654* | Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | 2cprodeq2dv 14655* | Equality deduction for double product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq12dv 14656* | Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodeq12rdv 14657* | Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prod2id 14658* | The second class argument to a product can be chosen so that it is always a set. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodrblem 14659* | Lemma for prodrb 14662. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | fprodcvg 14660* | The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodrblem2 14661* | Lemma for prodrb 14662. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodrb 14662* | Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodmolem3 14663* | Lemma for prodmo 14666. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodmolem2a 14664* | Lemma for prodmo 14666. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodmolem2 14665* | Lemma for prodmo 14666. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | prodmo 14666* | A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.) |
Theorem | zprod 14667* | Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.) |
Theorem | iprod 14668* | Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.) |
Theorem | zprodn0 14669* | Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.) |
Theorem | iprodn0 14670* | Nonzero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.) |
Theorem | fprod 14671* | The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) |
Theorem | fprodntriv 14672* | A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.) |
Theorem | prod0 14673 | A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.) |
Theorem | prod1 14674* | Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) |
Theorem | prodfc 14675* | A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.) |
Theorem | fprodf1o 14676* | Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.) |
Theorem | prodss 14677* | Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) |
Theorem | fprodss 14678* | Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.) |
Theorem | fprodser 14679* | A finite product expressed in terms of a partial product of an infinite sequence. The recursive definition of a finite product follows from here. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodcl2lem 14680* | Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodcllem 14681* | Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodcl 14682* | Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodrecl 14683* | Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodzcl 14684* | Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodnncl 14685* | Closure of a finite product of positive integers. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodrpcl 14686* | Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodnn0cl 14687* | Closure of a finite product of nonnegative integers. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprodcllemf 14688* | Finite product closure lemma. A version of fprodcllem 14681 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Theorem | fprodreclf 14689* | Closure of a finite product of real numbers. A version of fprodrecl 14683 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Theorem | fprodmul 14690* | The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fproddiv 14691* | The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.) |
Theorem | prodsn 14692* | A product of a singleton is the term. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | fprod1 14693* | A finite product of only one term is the term itself. (Contributed by Scott Fenton, 14-Dec-2017.) |
Theorem | prodsnf 14694* | A product of a singleton is the term. A version of prodsn 14692 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Theorem | climprod1 14695 | The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017.) |
Theorem | fprodsplit 14696* | Split a finite product into two parts. (Contributed by Scott Fenton, 16-Dec-2017.) |
Theorem | fprodm1 14697* | Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.) |
Theorem | fprod1p 14698* | Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.) |
Theorem | fprodp1 14699* | Multiply in the last term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.) |
Theorem | fprodm1s 14700* | Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.) |
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