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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgeoser 12601* The value of the finite geometric series ... . (Contributed by NM, 12-May-2006.) (Proof shortened by Mario Carneiro, 15-Jun-2014.)

Theoremgeolim 12602* The partial sums in the infinite series ... converge to . (Contributed by NM, 15-May-2006.)

Theoremgeolim2 12603* The partial sums in the geometric series ... converge to . (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeoreclim 12604* The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeo2sum 12605* 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.)

Theoremgeo2sum2 12606* The value of the finite geometric series ... . (Contributed by Mario Carneiro, 7-Sep-2016.)
..^

Theoremgeo2lim 12607* The value of the infinite geometric series ... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremgeomulcvg 12608* The geometric series converges even if it is multiplied by to result in the larger series . (Contributed by Mario Carneiro, 27-Mar-2015.)

Theoremgeoisum 12609* The infinite sum of ... is . (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeoisumr 12610* The infinite sum of reciprocals ... is . (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeoisum1 12611* The infinite sum of ... is . (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeoisum1c 12612* The infinite sum of ... is . (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theorem0.999... 12613 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.)

Theoremgeoihalfsum 12614 Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 12611. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 12613 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.)

5.8.10  Ratio test for infinite series convergence

Theoremcvgrat 12615* 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.)

5.8.11  Mertens' theorem

Theoremmertenslem1 12616* Lemma for mertens 12618. (Contributed by Mario Carneiro, 29-Apr-2014.)

Theoremmertenslem2 12617* Lemma for mertens 12618. (Contributed by Mario Carneiro, 28-Apr-2014.)

Theoremmertens 12618* Mertens' thoerem. 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.)

5.9  Elementary trigonometry

5.9.1  The exponential, sine, and cosine functions

Syntaxce 12619 Extend class notation to include the exponential function.

Syntaxceu 12620 Extend class notation to include Euler's constant = 2.7182818....

Syntaxcsin 12621 Extend class notation to include the sine function.

Syntaxccos 12622 Extend class notation to include the cosine function.

Syntaxctan 12623 Extend class notation to include the tangent function.

Syntaxcpi 12624 Extend class notation to include pi = 3.14159....

Definitiondf-ef 12625* Define the exponential function. (Contributed by NM, 14-Mar-2005.)

Definitiondf-e 12626 Define Euler's constant 2.71828.... (Contributed by NM, 14-Mar-2005.)

Definitiondf-sin 12627 Define the sine function. (Contributed by NM, 14-Mar-2005.)

Definitiondf-cos 12628 Define the cosine function. (Contributed by NM, 14-Mar-2005.)

Definitiondf-tan 12629 Define the tangent function. We define it this way for cmpt 4226, which requires the form . (Contributed by Mario Carneiro, 14-Mar-2014.)

Definitiondf-pi 12630 Define pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (We use the inverse of less-than, " ", to turn supremum into infimum; currently we don't have infimum defined separately.) (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremeftcl 12631 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)

Theoremreeftcl 12632 The terms of the series expansion of the exponential function of a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)

Theoremeftabs 12633 The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)

Theoremeftval 12634* The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefcllem 12635* Lemma for efcl 12640. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 12615 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)

Theoremef0lem 12636* The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefval 12637* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremesum 12638 Value of Euler's constant = 2.71828... (Contributed by Steve Rodriguez, 5-Mar-2006.)

Theoremeff 12639 Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)

Theoremefcl 12640 Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremefval2 12641* Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)

Theoremefcvg 12642* The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefcvgfsum 12643* Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremreefcl 12644 The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremreefcld 12645 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremere 12646 Euler's constant = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)

Theoremege2le3 12647 Lemma for egt2lt3 12760. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)

Theoremef0 12648 Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefcj 12649 Exponential function of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefadd 12651 Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)

Theoremefcan 12652 Cancellation of law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)

Theoremefne0 12653 The exponential function never vanishes. Corollary 15-4.3 of [Gleason] p. 309. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremefneg 12654 Exponent of a negative number. (Contributed by Mario Carneiro, 10-May-2014.)

Theoremeff2 12655 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)

Theoremefsub 12656 Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremefexp 12657 Exponential function to an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremefzval 12658 Value of the exponential function for integers. Special case of efval 12637. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremefgt0 12659 The exponential function of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpefcl 12660 The exponential function of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)

Theoremrpefcld 12661 The exponential function of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremeftlcvg 12662* The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014.)

Theoremeftlcl 12663* Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremreeftlcl 12664* Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremeftlub 12665* An upper bound on the absolute value of the infinite tail of the series expansion of the exponential function on the closed unit disk. (Contributed by Paul Chapman, 19-Jan-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)

Theoremefsep 12666* Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremeffsumlt 12667* The partial sums of the series expansion of the exponential function of a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremeft0val 12668 The value of the first term of the series expansion of the exponential function is 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremef4p 12669* Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremefgt1p2 12670 The exponential function of a positive real number is greater than the first three terms of the series expansion. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremefgt1p 12671 The exponential function of a positive real number is greater than 1 plus that number. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremefgt1 12672 The exponential function of a positive real number is greater than 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremeflt 12673 The exponential function on the reals is strictly monotonic. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 17-Jul-2014.)

Theoremefle 12674 The exponential function on the reals is strictly monotonic. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremreef11 12675 The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Mario Carneiro, 11-Mar-2014.)

Theoremreeff1 12676 The exponential function maps real arguments one-to-one to positive reals. (Contributed by Steve Rodriguez, 25-Aug-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremeflegeo 12677 The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)

Theoremsinval 12678 Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremcosval 12679 Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremsinf 12680 Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremcosf 12681 Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremsincl 12682 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremcoscl 12683 Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremtanval 12684 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.)

Theoremtancl 12685 The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremsincld 12686 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcoscld 12687 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremtancld 12688 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremtanval2 12689 Express the tangent function directly in terms of . (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremtanval3 12690 Express the tangent function directly in terms of . (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremresinval 12691 The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)

Theoremrecosval 12692 The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)

Theoremefi4p 12693* Separate out the first four terms of the infinite series expansion of the exponential function of an imaginary number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremresin4p 12694* Separate out the first four terms of the infinite series expansion of the sine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrecos4p 12695* Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremresincl 12696 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)

Theoremrecoscl 12697 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)

Theoremretancl 12698 The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremresincld 12699 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrecoscld 12700 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)

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