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Theorem List for Metamath Proof Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisercolllem3 12401* Lemma for isercoll 12402. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremisercoll 12402* Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremisercoll2 12403* Generalize isercoll 12402 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremclimsup 12404* A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)

Theoremclimcau 12405* A converging sequence of complex numbers is a Cauchy sequence. Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremclimbdd 12406* A converging sequence of complex numbers is bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremcaucvgrlem 12407* Lemma for caurcvgr 12408. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremcaurcvgr 12408* A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremcaucvgrlem2 12409* Lemma for caucvgr 12410. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.)

Theoremcaucvgr 12410* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremcaurcvg 12411* A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremcaurcvg2 12412* A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.)

Theoremcaucvg 12413* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremcaucvgb 12414* A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremserf0 12415* If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)

Theoremiseraltlem1 12416* Lemma for iseralt 12419. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremiseraltlem2 12417* Lemma for iseralt 12419. The terms of an alternating series form a chain of inequalities in alternate terms, so that for example and (assuming so that these terms are defined). (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremiseraltlem3 12418* Lemma for iseralt 12419. From iseraltlem2 12417, we have and , and we also have for each by the definition of the partial sum , so combining the inequalities we get , so and . Thus, both even and odd partial sums are Cauchy if converges to . (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremiseralt 12419* The alternating series test. If is a decreasing sequence that converges to , then is a convergent series. (Note that the first term is positive if is even, and negative if is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by using isermulc2 12392.) (Contributed by Mario Carneiro, 7-Apr-2015.)

5.8.3  Finite and infinite sums

Syntaxcsu 12420 Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)

Definitiondf-sum 12421* Define the sum of a series with an index set of integers . is normally a free variable in , i.e. can be thought of as . This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e. finite subsets of the upper integers) by summo 12452. Examples: means , and means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12600). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumex 12422 A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumeq1f 12423 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumeq1 12424* Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremnfsum1 12425* Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremnfsum 12426* Bound-variable hypothesis builder for sum: if is (effectively) not free in and , it is not free in . (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumeq2w 12427* Equality theorem for sum, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremsumeq2ii 12428* Equality theorem for sum, with the class expressions and guarded by to be always sets. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremsumeq2 12429* Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremcbvsum 12430* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremcbvsumv 12431* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremcbvsumi 12432* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)

Theoremsumeq1i 12433* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)

Theoremsumeq2i 12434* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)

Theoremsumeq12i 12435* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)

Theoremsumeq1d 12436* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)

Theoremsumeq2d 12437* Equality deduction for sum. Note that unlike sumeq2dv 12438, may occur in . (Contributed by NM, 1-Nov-2005.)

Theoremsumeq2dv 12438* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremsumeq2sdv 12439* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)

Theorem2sumeq2dv 12440* Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremsumeq12dv 12441* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)

Theoremsumeq12rdv 12442* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)

Theoremsum2id 12443* The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumfc 12444* A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremfz1f1o 12445* A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)

Theoremsumrblem 12446* Lemma for sumrb 12448. (Contributed by Mario Carneiro, 12-Aug-2013.)

Theoremfsumcvg 12447* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)

Theoremsumrb 12448* Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.)

Theoremsummolem3 12449* Lemma for summo 12452. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremsummolem2a 12450* Lemma for summo 12452. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 20-Apr-2014.)

Theoremsummolem2 12451* Lemma for summo 12452. (Contributed by Mario Carneiro, 3-Apr-2014.)

Theoremsummo 12452* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremzsum 12453* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 9-Apr-2014.)

Theoremisum 12454* Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)

Theoremfsum 12455* The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremsum0 12456 Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)

Theoremsumz 12457* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)

Theoremfsumf1o 12458* Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)

Theoremsumss 12459* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.)

Theoremfsumss 12460* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)

Theoremsumss2 12461* Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)

Theoremfsumcvg2 12462* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)

Theoremfsumsers 12463* Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 21-Apr-2014.)

Theoremfsumcvg3 12464* A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremfsumser 12465* A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 12476 and fsump1i 12494, which should make our notation clear and from which, along with closure fsumcl 12468, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)

Theoremfsumcl2lem 12466* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.)

Theoremfsumcllem 12467* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.)

Theoremfsumcl 12468* Closure of a finite sum of complex numbers . (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)

Theoremfsumrecl 12469* Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)

Theoremfsumzcl 12470* Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)

Theoremfsumnn0cl 12471* Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremfsumrpcl 12472* Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.)

Theoremfsumadd 12473* The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)

Theoremfsumsplit 12474* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)

Theoremsumsn 12475* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)

Theoremfsum1 12476* The finite sum of from to (i.e. a sum with only one term) is i.e. . (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)

Theoremsumsns 12477* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)

Theoremfsumm1 12478* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)

Theoremfzosump1 12479* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.)
..^ ..^

Theoremfsum1p 12480* Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremfsump1 12481* The addition of the next term in a finite sum of is the current term plus i.e. . (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)

Theoremisumclim 12482* An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremisumclim2 12483* A converging series converges to its infinite sum. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremisumclim3 12484* The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that must not occur in . (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremsumnul 12485* The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 4-Nov-2007.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremisumcl 12486* The sum of a converging infinite series is a complex number. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremisummulc2 12487* An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremisummulc1 12488* An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremisumdivc 12489* An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremisumrecl 12490* The sum of a converging infinite real series is a real number. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremisumge0 12491* An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.)

Theoremisumadd 12492* Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremsumsplit 12493* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremfsump1i 12494* Optimized version of fsump1 12481 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfsum2dlem 12495* Lemma for fsum2d 12496- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfsum2d 12496* Write a double sum as a sum over a two-dimensional region. Note that is a function of . (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremfsumxp 12497* Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfsumcnv 12498* Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfsumcom2 12499* Interchange order of summation. Note that and are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)

Theoremfsumcom 12500* Interchange order of summation. (Contributed by NM, 15-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)

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