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  • contraharmonic proportion | planetmath.org
    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 3A11Z05 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 3A11A05 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 3A11A05 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 Three positive numbers x 𝑥 m 𝑚 y 𝑦 are in contraharmonic proportion if the ratio of the difference of the second and the first number to the difference of the third and the second number is equal the ratio of the third and the first number i e if m x y m y x 𝑚 𝑥 𝑦 𝑚 𝑦 𝑥 1 The middle number m 𝑚 is then called the contraharmonic mean sometimes antiharmonic mean of the first and the last number The contraharmonic proportion has very probably been

    Original URL path: http://www.planetmath.org/contraharmonicproportion (2016-04-25)
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  • convergence condition of infinite product | planetmath.org
    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 3A01A17 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 3A01A17 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 3A01A20 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 3A01A20 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang

    Original URL path: http://www.planetmath.org/convergenceconditionofinfiniteproduct (2016-04-25)
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  • convergence of complex term series | planetmath.org
    ν a ν b ν ℝ ν fragments subscript c ν subscript a ν i subscript b ν italic fragments normal subscript a ν normal subscript b ν R for all ν normal c nu a nu ib nu qquad a nu b nu in mathbb R forall nu is convergent iff the sequence of its partial sums converges to a complex number Theorem 1 The series 1 converges iff the series ν 1 a ν and ν 1 b ν superscript subscript ν 1 subscript a ν and superscript subscript ν 1 subscript b ν displaystyle sum nu 1 infty a nu quad mbox and quad sum nu 1 infty b nu 2 formed by real parts and the imaginary parts of its terms both are convergent Proof Let ε 0 ε 0 varepsilon 0 Denote ν 1 n a ν s n ν 1 n b ν t n ν 1 n c ν u n formulae sequence assign superscript subscript ν 1 n subscript a ν subscript s n formulae sequence assign superscript subscript ν 1 n subscript b ν subscript t n assign superscript subscript ν 1 n subscript c ν subscript u n sum nu 1 n a nu s n quad sum nu 1 n b nu t n quad sum nu 1 n c nu u n If the series 2 are convergent with sums S S S and T T T then there is a number N N N such that s n S ε 2 t n T ε 2 when n N formulae sequence subscript s n S ε 2 formulae sequence subscript t n T ε 2 when n N s n S frac varepsilon 2 quad t n T frac varepsilon 2 quad mbox when quad n geqq N Accordingly u n S i T s n S 2 t n T 2 s n S t n T ε when n N formulae sequence subscript u n S i T superscript subscript s n S 2 superscript subscript t n T 2 subscript s n S subscript t n T ε when n N u n S iT sqrt s n S 2 t n T 2 leqq s n S t n T varepsilon quad mbox when quad n geqq N i e the series 1 converges to S i T S i T S iT If conversely 1 converges to a complex number u s i t s t ℝ fragments u s i t fragments normal s normal t R normal normal u s it quad s t in mathbb R then s n s s n s i t n t u n u t n t s n s i t n t u n u formulae sequence subscript s n s subscript s n s i subscript t n t subscript u n u subscript t n t subscript s n s i subscript t n t subscript u n u

    Original URL path: http://www.planetmath.org/convergenceofcomplextermseries (2016-04-25)
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  • convergence of integrals | planetmath.org
    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 convergence of integrals Similarly as one speaks of convergence of series one can speak of convergence of integrals especially of Riemann integrals I f t d t subscript I f t d t int I f t dt This integral is convergent if it exists and otherwise divergent One can also speak of absolute convergence of integrals Example Study the convergence of the integral 1 2 d x ln x c superscript subscript 1 2 d x superscript x c displaystyle int 1 2 frac dx ln x c 1 where c c c is a real constant According to the logarithm series we may write for 1 x b 1 x b 1 x b where b b b is sufficiently close to 1 1 1 the estimations ln x 1 x 1 O x 1 2 x 1 1 O x 1 2 x 1 1 2 x 1 x 1 x 1 O superscript x 1 2 x 1 1 O x 1 2 x1 12 x1 displaystyle ln x 1 x 1 O x 1 2 x 1 1 O x 1 begin cases leq 2 x 1 geq frac 1 2 x 1 end cases Let 1 a b 1 a b 1 a b 1 superscript 1 1 circ For c 1 c 1 c 1 a b d x ln x c superscript subscript a b d x superscript x c displaystyle int a b frac dx ln x c a b d x 2 c x 1 c 1 2 c x a b 1 c 1 x 1 c 1 fragments superscript subscript a b d x superscript 2 c superscript x 1 c 1 superscript 2 c superscript subscript normal x a b 1 c 1 superscript x 1 c 1 displaystyle geqq int a b frac dx 2 c x 1 c frac 1 2 c operatornamewithlimits Big x a quad b frac 1 c 1 x 1 c 1 1 2 c c 1 1 a 1 c 1 1 b 1 c 1 as a 1 fragments 1 superscript 2 c c 1 fragments normal 1 superscript a 1 c 1 1 superscript b 1 c 1 normal normal as a normal 1 displaystyle frac 1 2 c c 1 left frac 1 a 1 c 1 frac 1 b 1 c 1 right longrightarrow infty quad mbox as quad a to 1 2 superscript 2 2 circ For c 1 c 1 c 1 a b d x ln x superscript subscript a b d x x displaystyle int a b frac dx ln x

    Original URL path: http://www.planetmath.org/convergenceofintegrals (2016-04-25)
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  • convergence of Riemann zeta series | planetmath.org
    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 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 The series n 1 1 n s superscript subscript 𝑛 1 1 superscript 𝑛 𝑠 1 converges absolutely for all s 𝑠 with real part greater than 1 Proof Let s σ i t 𝑠 𝜎 𝑖 𝑡 where σ 𝜎 and t 𝑡 are real numbers and σ 1 𝜎 1 Then 1 n s 1 e s log n 1 e σ log n 1 n σ 1 superscript 𝑛 𝑠 1 superscript 𝑒 𝑠 𝑛 1 superscript 𝑒 𝜎 𝑛 1 superscript 𝑛 𝜎 Since the series n 1 1 n σ superscript subscript 𝑛 1 1 superscript 𝑛 𝜎 converges by the p 𝑝 test for σ 1 𝜎 1 we conclude that the series 1 is

    Original URL path: http://www.planetmath.org/convergenceofriemannzetaseries (2016-04-25)
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  • convergent series where not only$~a_n$ but also $na_n$ tends to 0 | planetmath.org
    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 3A30A99 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 convergent series where not only a n subscript a n a n but also n a n n subscript a n na n tends to 0 Proposition If the terms a n subscript a n a n of the convergent series a 1 a 2 subscript a 1 subscript a 2 normal a 1 a 2 ldots are positive and form a monotonically decreasing sequence then lim n n a n 0 subscript normal n n subscript a n 0 displaystyle lim n to infty na n 0 1 Proof Let ε ε varepsilon be any positive number By the Cauchy criterion for convergence and the positivity of the terms there is a positive integer m m m such that 0 a m 1 a m p ε 2 p 1 2 fragments 0 subscript a m 1 normal subscript a m p ε 2 italic fragments normal p 1 normal 2 normal normal normal normal 0 a m 1 ldots a m p frac varepsilon 2 qquad p 1 2 ldots Since the sequence a 1 a 2 subscript a 1 subscript a 2 normal a 1 a 2 ldots is decreasing this implies 0 p a m p ε 2 p 1 2 fragments 0 p subscript a m p ε 2 italic fragments normal p 1 normal 2 normal normal normal normal displaystyle 0 pa m p frac varepsilon 2 qquad p 1 2 ldots 2 Choosing here especially p m assign p m p m we get 0 m a m m ε 2 0 m subscript a m m ε 2 0 ma m m frac varepsilon 2 whence again due to the decrease 0 m a m p ε 2 p m m 1 fragments 0 m subscript a m p ε 2 italic fragments normal p m normal m 1 normal normal normal normal displaystyle 0 ma m p frac varepsilon 2 qquad p m m 1 ldots 3 Adding the inequalities 2 and 3 with the common values p m m 1 p m m 1 normal p m m 1 ldots then yields 0 m p a m p ε for p m formulae sequence 0 m p subscript a m p ε for p m 0 m p a m p varepsilon qquad mbox for quad p geqq m This may be

    Original URL path: http://www.planetmath.org/convergentserieswherenotonlyanbutalsonantendsto0 (2016-04-25)
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  • converging alternating series not satisfying all Leibniz' conditions | planetmath.org
    satisfying all Leibniz conditions The alternating series n 1 1 n 1 n 1 n 1 1 2 1 1 1 4 1 3 1 6 1 5 fragments superscript subscript n 1 superscript 1 n 1 n superscript 1 n 1 1 2 1 1 1 4 1 3 1 6 1 5 normal displaystyle sum n 1 infty frac 1 n 1 n 1 n 1 frac 1 2 frac 1 1 frac 1 4 frac 1 3 frac 1 6 frac 1 5 ldots 1 satisfies the other requirements of Leibniz test except the monotonicity of the absolute values of the terms The convergence may however be shown by manipulating the terms as follows We first multiply the numerator and the denominator of the general term by the difference n 1 n 1 n superscript 1 n 1 n 1 n 1 getting from 1 n 1 1 n 1 n 1 n 1 1 2 n 2 n 1 n 1 n 2 1 1 n 1 1 2 n 2 1 n 1 n n 2 1 1 n 2 1 superscript subscript n 1 superscript 1 n 1 n superscript 1 n 1 1 2 superscript subscript n 2 n superscript 1 n 1 superscript n 2 1 superscript 1 n 1 1 2 superscript subscript n 2 superscript 1 n 1 n superscript n 2 1 1 superscript n 2 1 displaystyle sum n 1 infty frac 1 n 1 n 1 n 1 frac 1 2 sum n 2 infty frac n 1 n 1 n 2 1 1 n 1 frac 1 2 sum n 2 infty left frac 1 n 1 n n 2 1 frac 1 n 2 1 right 2 One can state that the series n 2

    Original URL path: http://www.planetmath.org/convergingalternatingseriesnotsatisfyingallleibnizconditions (2016-04-25)
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  • converse | planetmath.org
    3E 7B msc 3A40 00 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 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 converse Let a statement be of the form of an implication If p p p then q q q i e it has a certain premise p p p and a conclusion q q q The statement in which one has interchanged the conclusion and the premise If q q q then p p p is the converse of the first In other words from the former one concludes that q q q is necessary for p p p and from the latter that p p p is necessary for q q q Note that the converse of an implication and the inverse of the same implication are contrapositives of each other and thus are logically equivalent If there is originally a statement which is a true theorem and if its converse also is true then the latter can be called the converse theorem of the original one Note that if the converse of a true theorem If p p p then q q q is also true then p p p iff q q q is a true theorem For example we know the theorem on isosceles triangles If

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