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  • a series related to harmonic series | planetmath.org
    error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A15 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A15 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A03 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A03 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22

    Original URL path: http://www.planetmath.org/aseriesrelatedtoharmonicseries (2016-04-25)
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  • a special case of partial integration | planetmath.org
    of partial integration In determining the antiderivative of a transcendental function U U U whose derivative U superscript U normal U prime is algebraic the result can be obtained when choosing in the formula U V d x U V V U d x U superscript V normal d x U V V superscript U normal d x int UV prime dx UV int VU prime dx of integration by parts V 1 superscript V normal 1 V prime equiv 1 then one has U d x U 1 d x U x x U d x U d x normal U 1 d x normal U x normal x superscript U normal d x int U dx int U cdot 1 dx U cdot x int x cdot U prime dx The functions U U U in question are mainly the logarithm the cyclometric functions and the area functions Examples 1 ln x d x x ln x x 1 x d x x ln x x C x d x x x normal x 1 x d x x x x C displaystyle int ln x dx x ln x int x cdot frac 1 x dx x ln x x C 2 arcsin x d x x arcsin x x 1 1 x 2 d x x arcsin x 1 2 2 x 1 x 2 d x x arcsin x 1 x 2 C x d x x x normal x 1 1 superscript x 2 d x x x 1 2 2 x 1 superscript x 2 d x x x 1 superscript x 2 C displaystyle int arcsin x dx x arcsin x int x cdot frac 1 sqrt 1 x 2 dx x arcsin x frac 1 2 int frac 2x sqrt 1 x 2 dx x arcsin x sqrt 1 x 2 C 3 arctan x d x x arctan x x 1 1 x 2 d x x arctan x 1 2 2 x 1 x 2 d x x arctan x 1 2 ln 1 x 2 C x arctan x ln 1 x 2 C x d x x x normal x 1 1 superscript x 2 d x x x 1 2 2 x 1 superscript x 2 d x x x 1 2 1 superscript x 2 C x x 1 superscript x 2 C displaystyle int arctan x dx x arctan x int x cdot frac 1 1 x 2 dx x arctan x frac 1 2 int frac 2x 1 x 2 dx x arctan x frac 1 2 ln 1 x 2 C x arctan x ln sqrt 1 x 2 C 4 arcosh x d x x arcosh x x 1 x 2 1 d x x arcosh x x 2 1 C arcosh x d x x arcosh x normal x 1 superscript x 2 1 d x x arcosh x superscript x 2 1 C displaystyle int arcosh x dx x

    Original URL path: http://www.planetmath.org/aspecialcaseofpartialintegration (2016-04-25)
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  • a sufficient condition for convergence of integral | planetmath.org
    f is positive and continuous on the interval a a a infty A sufficient condition for the convergence of the improper integral a f x d x superscript subscript a f x d x displaystyle int a infty f x dx 1 is that lim x f x 1 f x q 1 subscript normal x f x 1 f x q 1 displaystyle lim x to infty frac f x 1 f x q 1 2 Proof Assume that the condition 2 is in force For an indirect proof make the antithesis that the integral 1 diverges Because of the positiveness we have a f x d x superscript subscript a f x d x int a infty f x dx infty We can use l Hôpital s rule lim c a c f x 1 d x a c f x d x lim c f c 1 f c subscript normal c superscript subscript a c f x 1 d x superscript subscript a c f x d x subscript normal c f c 1 f c lim c to infty frac int a c f x 1 dx int a c f x dx lim c to infty frac f c 1 f c Using the substitution x 1 t x 1 t x 1 t we get a c f x 1 d x a 1 c 1 f t d t a 1 a f t d t a c f t d t c 1 c f t d t superscript subscript a c f x 1 d x superscript subscript a 1 c 1 f t d t superscript subscript a 1 a f t d t superscript subscript a c f t d t superscript subscript c 1 c f t

    Original URL path: http://www.planetmath.org/asufficientconditionforconvergenceofintegral (2016-04-25)
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  • Abel's multiplication rule for series | planetmath.org
    tabs View active tab Coauthors PDF Source Edit Cauchy has originally presented the multiplication rule j 1 a j k 1 b k n 1 a 1 b n a 2 b n 1 a n b 1 superscript subscript j 1 normal subscript a j superscript subscript k 1 subscript b k superscript subscript n 1 subscript a 1 subscript b n subscript a 2 subscript b n 1 normal subscript a n subscript b 1 displaystyle sum j 1 infty a j cdot sum k 1 infty b k sum n 1 infty a 1 b n a 2 b n 1 ldots a n b 1 1 for two series His assumption was that both of the multiplicand series should be absolutely convergent Mertens 1875 lightened the assumption requiring that both multiplicands should be convergent but at least one of them absolutely convergent see the parent entry N H Abel s most general form of the multiplication rule is the Theorem The rule 1 for multiplication of series with real or complex terms is valid as soon as all three of its series are convergent Proof We consider the corresponding power series j 1 a j x j k 1 b k x k superscript subscript j 1 subscript a j superscript x j superscript subscript k 1 subscript b k superscript x k displaystyle sum j 1 infty a j x j qquad sum k 1 infty b k x k 2 When x 1 x 1 x 1 they give the series j 1 a j k 1 b k superscript subscript j 1 subscript a j superscript subscript k 1 subscript b k sum j 1 infty a j quad sum k 1 infty b k which we assume to converge Thus the power series are absolutely convergent for x 1 x 1 x 1 whence they obey the multiplication rule due to Cauchy j 1 a j x j k 1 b k x k n 1 a 1 b n a 2 b n 1 a n b 1 x n 1 superscript subscript j 1 normal subscript a j superscript x j superscript subscript k 1 subscript b k superscript x k superscript subscript n 1 subscript a 1 subscript b n subscript a 2 subscript b n 1 normal subscript a n subscript b 1 superscript x n 1 displaystyle sum j 1 infty a j x j cdot sum k 1 infty b k x k sum n 1 infty a 1 b n a 2 b n 1 ldots a n b 1 x n 1 3 On the other hand the sums of the power series 2 are as is well known continuous functions on the interval 0 1 0 1 0 1 the same concerns the right hand side of 3 because for x 1 x 1 x 1 it becomes the third series which we assume convergent When x 1 normal x limit from 1

    Original URL path: http://www.planetmath.org/abelsmultiplicationruleforseries (2016-04-25)
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  • absolute convergence of double series | planetmath.org
    Primary tabs View active tab Coauthors PDF Source Edit absolute convergence of double series Let us consider the double series i j 1 u i j superscript subscript i j 1 subscript u i j displaystyle sum i j 1 infty u ij 1 of real or complex numbers u i j subscript u i j u ij Denote the row series u k 1 u k 2 subscript u k 1 subscript u k 2 normal u k1 u k2 ldots by R k subscript R k R k the column series u 1 k u 2 k subscript u 1 k subscript u 2 k normal u 1k u 2k ldots by C k subscript C k C k and the diagonal series u 11 u 12 u 21 u 13 u 22 u 31 subscript u 11 subscript u 12 subscript u 21 subscript u 13 subscript u 22 subscript u 31 normal u 11 u 12 u 21 u 13 u 22 u 31 ldots by DS Then one has the Theorem All row series all column series and the diagonal series converge absolutely and k 1 R k k 1 C k DS superscript subscript k 1 subscript R k superscript subscript k 1 subscript C k DS sum k 1 infty R k sum k 1 infty C k mbox DS if one of the following conditions is true The diagonal series converges absolutely There exists a positive number M M M such that every finite sum of the numbers u i j subscript u i j u ij is M absent M leqq M The row series R k subscript R k R k converge absolutely and the series W 1 W 2 subscript W 1 subscript W 2 normal W 1 W 2 ldots with terms j 1 u k j W k superscript subscript j 1 subscript u k j subscript W k sum j 1 infty u kj W k is convergent An analogical condition may be formulated for the column series C k subscript C k C k Example Does the double series m 2 n 3 n m superscript subscript m 2 superscript subscript n 3 superscript n m sum m 2 infty sum n 3 infty n m converge If yes determine its sum The column series m 2 1 n m superscript subscript m 2 superscript 1 n m displaystyle sum m 2 infty left frac 1 n right m have positive terms and are absolutely converging geometric series having the sum 1 n 2 1 1 n 1 n n 1 1 n 1 1 n W n superscript 1 n 2 1 1 n 1 n n 1 1 n 1 1 n subscript W n frac 1 n 2 1 1 n frac 1 n n 1 frac 1 n 1 frac 1 n W n The series W 3 W 4 subscript W 3 subscript W 4 normal W 3 W 4

    Original URL path: http://www.planetmath.org/absoluteconvergenceofdoubleseries (2016-04-25)
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  • absolute convergence of infinite product and series | planetmath.org
    2Frdf sink 23this 3E 7B msc 3A40A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A40A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit absolute convergence of infinite product and series Theorem The infinite product n 1 1 c n superscript subscript product n 1 1 subscript c n prod n 1 infty 1 c n converges absolutely if and only if the series n 1 c n superscript subscript n 1 subscript c n sum n 1 infty c n with complex terms c n subscript c n c n converges absolutely Proof The theorem follows directly from the theorems of the entries absolutely convergent infinite product converges and infinite product of sums 1 a i 1 subscript a i 1 a i Keywords absolute convergence necessary and sufficient

    Original URL path: http://www.planetmath.org/absoluteconvergenceofinfiniteproductandseries (2016-04-25)
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  • absolute convergence of integral and boundedness of derivative | planetmath.org
    infty f x dx where the real function f f f and its derivative f superscript f normal f prime are continuous and f superscript f normal f prime additionally bounded on the interval a a a infty Then lim x f x 0 subscript normal x f x 0 displaystyle lim x to infty f x 0 1 Proof If c a c a c a we obtain a c f x f x d x 1 2 a c f x 2 f c 2 f a 2 2 fragments superscript subscript a c f fragments normal x normal superscript f normal fragments normal x normal d x 1 2 superscript subscript normal a c superscript fragments normal f fragments normal x normal normal 2 superscript f c 2 superscript f a 2 2 normal int a c f x f prime x dx frac 1 2 operatornamewithlimits Big a quad c f x 2 frac f c 2 f a 2 2 from which f c 2 f a 2 2 a c f x f x d x superscript f c 2 superscript f a 2 2 superscript subscript a c f x superscript f normal x d x displaystyle f c 2 f a 2 2 int a c f x f prime x dx 2 Using the boundedness of f superscript f normal f prime and the absolute convergence we can estimate upwards the integral a c f x f x d x a c f x f x d x M a c f x d x M a f x d x c a formulae sequence superscript subscript a c f x superscript f normal x d x superscript subscript a c f x superscript f normal x d x M superscript subscript a c f x d x M superscript subscript a f x d x for all c a int a c f x f prime x dx int a c f x f prime x dx leqq M int a c f x dx leqq M int a infty f x dx quad forall c in a infty whence a f x f x d x superscript subscript a f x superscript f normal x d x int a infty f x f prime x dx is finite and thus a f x f x d x superscript subscript a f x superscript f normal x d x int a infty f x f prime x dx converges absolutely Hence 2 implies lim c f c 2 f a 2 2 a f x f x d x subscript normal c superscript f c 2 superscript f a 2 2 superscript subscript a f x superscript f normal x d x lim c to infty f c 2 f a 2 2 int a infty f x f prime x dx i e lim x f x 2 subscript normal x superscript f x 2 displaystyle lim x to

    Original URL path: http://www.planetmath.org/absoluteconvergenceofintegralandboundednessofderivative (2016-04-25)
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  • absolutely convergent infinite product converges | planetmath.org
    3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A40A10 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit absolutely convergent infinite product converges Theorem An absolutely convergent infinite product ν 1 1 c ν 1 c 1 1 c 2 1 c 3 superscript subscript product ν 1 1 subscript c ν 1 subscript c 1 1 subscript c 2 1 subscript c 3 normal displaystyle prod nu 1 infty 1 c nu 1 c 1 1 c 2 1 c 3 cdots 1 of complex numbers is convergent Proof We thus assume the convergence of the product ν 1 1 c ν 1 c 1 1 c 2 1 c 3 superscript subscript product ν 1 1 subscript c ν 1 subscript c 1 1 subscript c 2 1 subscript c 3 normal displaystyle prod nu 1 infty 1 c nu 1 c 1 1 c 2 1 c 3 cdots 2 Let ε ε varepsilon be an arbitrary positive number By the general convergence condition of infinite product we have 1 c n 1 1 c n 2 1 c n p 1 ε p ℤ formulae sequence 1 subscript c n 1 1 subscript c n 2 normal 1 subscript c n p 1 ε for all p subscript ℤ 1 c n 1 1 c n 2 cdots 1 c n p 1 varepsilon quad forall p in mathbb Z when n n absent n geqq certain n ε subscript n ε n varepsilon Then we see that 1 c n 1 1 c n 2 1 c n p 1 1 subscript c n 1 1 subscript c n 2 normal 1 subscript c n p 1 displaystyle 1 c n 1 1 c n 2 cdots 1 c n p 1 1 ν n 1 n p c ν μ ν c μ c ν c n 1 c n 2 c n p 1 absent 1 superscript subscript ν n 1 n p subscript c ν subscript μ ν subscript c μ subscript c ν normal subscript c n 1 subscript c n 2 normal subscript c n p 1 displaystyle 1 sum nu n 1 n p c nu sum mu nu c mu c nu ldots c n 1 c n 2 cdots c n p 1 1 ν n 1 n p c ν μ ν c μ c ν c n 1 c n 2 c n p 1 absent 1 superscript subscript ν n 1 n p subscript c ν subscript μ ν subscript c μ subscript c ν normal subscript c n 1 subscript c n 2 normal subscript c n p 1 displaystyle leqq 1 sum nu n 1 n p c nu sum mu nu c

    Original URL path: http://www.planetmath.org/absolutelyconvergentinfiniteproductconverges (2016-04-25)
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