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How can I know a value of this limit?


Limits in integrationEvaluation of a limitCan this limit be solved algebraically?How can i resolve this limit without L'Hopital's Rule?find this limit without l'hopital's ruleHow to find the value of $a$ and $b$ from this limit problem with or without L'Hopital's formula?How to do limit?Find the limit of the sequence $(4^n)/((2n)!)$How to solve this limit involving sine and log?Find a limit, no Taylor formula













1












$begingroup$


$$
lim_tto 1^- (1-t) sum_n=0^infty fract^n1+t^n
$$



I try to use L'hopital's rule but not successfully. Also I know that exist formula Sonine but I can't understand how I can use it.






real-analysis limits







share|cite|improve this question











$endgroup$







  • 2




    $begingroup$
    is it $lim_tto 1^-$ on the limit or simply $lim_tto 1$?
    $endgroup$
    – cand
    yesterday







  • 1




    $begingroup$
    also try reading meta.math.stackexchange.com/questions/5020/… to learn how mathjax work so peoples can understand you equation part better.
    $endgroup$
    – cand
    yesterday










  • $begingroup$
    took closest interpretation, please check
    $endgroup$
    – gt6989b
    yesterday










  • $begingroup$
    it is lim t→1−0
    $endgroup$
    – Andrey Komisarov
    yesterday






  • 1




    $begingroup$
    @cand: the sum diverges for $tgt1$.
    $endgroup$
    – robjohn♦
    5 hours ago















1












$begingroup$


$$
lim_tto 1^- (1-t) sum_n=0^infty fract^n1+t^n
$$



I try to use L'hopital's rule but not successfully. Also I know that exist formula Sonine but I can't understand how I can use it.






real-analysis limits







share|cite|improve this question











$endgroup$







  • 2




    $begingroup$
    is it $lim_tto 1^-$ on the limit or simply $lim_tto 1$?
    $endgroup$
    – cand
    yesterday







  • 1




    $begingroup$
    also try reading meta.math.stackexchange.com/questions/5020/… to learn how mathjax work so peoples can understand you equation part better.
    $endgroup$
    – cand
    yesterday










  • $begingroup$
    took closest interpretation, please check
    $endgroup$
    – gt6989b
    yesterday










  • $begingroup$
    it is lim t→1−0
    $endgroup$
    – Andrey Komisarov
    yesterday






  • 1




    $begingroup$
    @cand: the sum diverges for $tgt1$.
    $endgroup$
    – robjohn♦
    5 hours ago













1












1








1


1



$begingroup$


$$
lim_tto 1^- (1-t) sum_n=0^infty fract^n1+t^n
$$



I try to use L'hopital's rule but not successfully. Also I know that exist formula Sonine but I can't understand how I can use it.






real-analysis limits







share|cite|improve this question











$endgroup$




$$
lim_tto 1^- (1-t) sum_n=0^infty fract^n1+t^n
$$



I try to use L'hopital's rule but not successfully. Also I know that exist formula Sonine but I can't understand how I can use it.






real-analysis limits




real-analysis limits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 7 hours ago









robjohn♦

269k27309635




269k27309635










asked yesterday









Andrey KomisarovAndrey Komisarov

353




353







  • 2




    $begingroup$
    is it $lim_tto 1^-$ on the limit or simply $lim_tto 1$?
    $endgroup$
    – cand
    yesterday







  • 1




    $begingroup$
    also try reading meta.math.stackexchange.com/questions/5020/… to learn how mathjax work so peoples can understand you equation part better.
    $endgroup$
    – cand
    yesterday










  • $begingroup$
    took closest interpretation, please check
    $endgroup$
    – gt6989b
    yesterday










  • $begingroup$
    it is lim t→1−0
    $endgroup$
    – Andrey Komisarov
    yesterday






  • 1




    $begingroup$
    @cand: the sum diverges for $tgt1$.
    $endgroup$
    – robjohn♦
    5 hours ago












  • 2




    $begingroup$
    is it $lim_tto 1^-$ on the limit or simply $lim_tto 1$?
    $endgroup$
    – cand
    yesterday







  • 1




    $begingroup$
    also try reading meta.math.stackexchange.com/questions/5020/… to learn how mathjax work so peoples can understand you equation part better.
    $endgroup$
    – cand
    yesterday










  • $begingroup$
    took closest interpretation, please check
    $endgroup$
    – gt6989b
    yesterday










  • $begingroup$
    it is lim t→1−0
    $endgroup$
    – Andrey Komisarov
    yesterday






  • 1




    $begingroup$
    @cand: the sum diverges for $tgt1$.
    $endgroup$
    – robjohn♦
    5 hours ago







2




2




$begingroup$
is it $lim_tto 1^-$ on the limit or simply $lim_tto 1$?
$endgroup$
– cand
yesterday





$begingroup$
is it $lim_tto 1^-$ on the limit or simply $lim_tto 1$?
$endgroup$
– cand
yesterday





1




1




$begingroup$
also try reading meta.math.stackexchange.com/questions/5020/… to learn how mathjax work so peoples can understand you equation part better.
$endgroup$
– cand
yesterday




$begingroup$
also try reading meta.math.stackexchange.com/questions/5020/… to learn how mathjax work so peoples can understand you equation part better.
$endgroup$
– cand
yesterday












$begingroup$
took closest interpretation, please check
$endgroup$
– gt6989b
yesterday




$begingroup$
took closest interpretation, please check
$endgroup$
– gt6989b
yesterday












$begingroup$
it is lim t→1−0
$endgroup$
– Andrey Komisarov
yesterday




$begingroup$
it is lim t→1−0
$endgroup$
– Andrey Komisarov
yesterday




1




1




$begingroup$
@cand: the sum diverges for $tgt1$.
$endgroup$
– robjohn♦
5 hours ago




$begingroup$
@cand: the sum diverges for $tgt1$.
$endgroup$
– robjohn♦
5 hours ago










2 Answers
2






active

oldest

votes


















5












$begingroup$

Fix $tin(0,1)$. Let $f(x)=fract^x1+t^x$. Using area principle, we have that
$$sum_n=0^infty f(n)=int_0^infty f(x) dx+E,$$
where $|E|le f(0)=frac12$.



It remains to estimate the integral. Let $y=t^x$. Then the integral can be rewritten as
$$int_0^infty f(x) dx=int_0^infty fract^x1+t^x dx=frac-1log tint_0^1fracdy1+y=frac-log2log t.$$
So the limit we are looking for is
$$lim_trightarrow1^-(1-t)sum_n=0^infty f(n)=lim_trightarrow1^-fract-1log tlog2=log2.$$






share|cite|improve this answer









$endgroup$




















    3












    $begingroup$

    For any $m$,
    $$
    beginalign
    (1-t)sum_n=0^inftyfract^n1+t^n
    &=(1-t)sum_n=0^inftysum_k=1^m-1(-1)^k-1t^kn\
    &+(-1)^m-1(1-t)sum_n=0^inftyfract^mn1+t^n\
    &=sum_k=1^m-1(-1)^k-1frac1-t1-t^k+O!left(frac1-t1-t^mright)tag1
    endalign
    $$
    Let $tto1^-$,
    $$
    lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
    =sum_k=1^m-1frac(-1)^k-1k+O!left(frac1mright)tag2
    $$
    Let $mtoinfty$,
    $$
    beginalign
    lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
    &=sum_k=1^inftyfrac(-1)^k-1k\[6pt]
    &=log(2)tag3
    endalign
    $$






    share|cite|improve this answer











    $endgroup$












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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      5












      $begingroup$

      Fix $tin(0,1)$. Let $f(x)=fract^x1+t^x$. Using area principle, we have that
      $$sum_n=0^infty f(n)=int_0^infty f(x) dx+E,$$
      where $|E|le f(0)=frac12$.



      It remains to estimate the integral. Let $y=t^x$. Then the integral can be rewritten as
      $$int_0^infty f(x) dx=int_0^infty fract^x1+t^x dx=frac-1log tint_0^1fracdy1+y=frac-log2log t.$$
      So the limit we are looking for is
      $$lim_trightarrow1^-(1-t)sum_n=0^infty f(n)=lim_trightarrow1^-fract-1log tlog2=log2.$$






      share|cite|improve this answer









      $endgroup$

















        5












        $begingroup$

        Fix $tin(0,1)$. Let $f(x)=fract^x1+t^x$. Using area principle, we have that
        $$sum_n=0^infty f(n)=int_0^infty f(x) dx+E,$$
        where $|E|le f(0)=frac12$.



        It remains to estimate the integral. Let $y=t^x$. Then the integral can be rewritten as
        $$int_0^infty f(x) dx=int_0^infty fract^x1+t^x dx=frac-1log tint_0^1fracdy1+y=frac-log2log t.$$
        So the limit we are looking for is
        $$lim_trightarrow1^-(1-t)sum_n=0^infty f(n)=lim_trightarrow1^-fract-1log tlog2=log2.$$






        share|cite|improve this answer









        $endgroup$















          5












          5








          5





          $begingroup$

          Fix $tin(0,1)$. Let $f(x)=fract^x1+t^x$. Using area principle, we have that
          $$sum_n=0^infty f(n)=int_0^infty f(x) dx+E,$$
          where $|E|le f(0)=frac12$.



          It remains to estimate the integral. Let $y=t^x$. Then the integral can be rewritten as
          $$int_0^infty f(x) dx=int_0^infty fract^x1+t^x dx=frac-1log tint_0^1fracdy1+y=frac-log2log t.$$
          So the limit we are looking for is
          $$lim_trightarrow1^-(1-t)sum_n=0^infty f(n)=lim_trightarrow1^-fract-1log tlog2=log2.$$






          share|cite|improve this answer









          $endgroup$



          Fix $tin(0,1)$. Let $f(x)=fract^x1+t^x$. Using area principle, we have that
          $$sum_n=0^infty f(n)=int_0^infty f(x) dx+E,$$
          where $|E|le f(0)=frac12$.



          It remains to estimate the integral. Let $y=t^x$. Then the integral can be rewritten as
          $$int_0^infty f(x) dx=int_0^infty fract^x1+t^x dx=frac-1log tint_0^1fracdy1+y=frac-log2log t.$$
          So the limit we are looking for is
          $$lim_trightarrow1^-(1-t)sum_n=0^infty f(n)=lim_trightarrow1^-fract-1log tlog2=log2.$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 19 hours ago









          Eric YauEric Yau

          48429




          48429





















              3












              $begingroup$

              For any $m$,
              $$
              beginalign
              (1-t)sum_n=0^inftyfract^n1+t^n
              &=(1-t)sum_n=0^inftysum_k=1^m-1(-1)^k-1t^kn\
              &+(-1)^m-1(1-t)sum_n=0^inftyfract^mn1+t^n\
              &=sum_k=1^m-1(-1)^k-1frac1-t1-t^k+O!left(frac1-t1-t^mright)tag1
              endalign
              $$
              Let $tto1^-$,
              $$
              lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
              =sum_k=1^m-1frac(-1)^k-1k+O!left(frac1mright)tag2
              $$
              Let $mtoinfty$,
              $$
              beginalign
              lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
              &=sum_k=1^inftyfrac(-1)^k-1k\[6pt]
              &=log(2)tag3
              endalign
              $$






              share|cite|improve this answer











              $endgroup$

















                3












                $begingroup$

                For any $m$,
                $$
                beginalign
                (1-t)sum_n=0^inftyfract^n1+t^n
                &=(1-t)sum_n=0^inftysum_k=1^m-1(-1)^k-1t^kn\
                &+(-1)^m-1(1-t)sum_n=0^inftyfract^mn1+t^n\
                &=sum_k=1^m-1(-1)^k-1frac1-t1-t^k+O!left(frac1-t1-t^mright)tag1
                endalign
                $$
                Let $tto1^-$,
                $$
                lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
                =sum_k=1^m-1frac(-1)^k-1k+O!left(frac1mright)tag2
                $$
                Let $mtoinfty$,
                $$
                beginalign
                lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
                &=sum_k=1^inftyfrac(-1)^k-1k\[6pt]
                &=log(2)tag3
                endalign
                $$






                share|cite|improve this answer











                $endgroup$















                  3












                  3








                  3





                  $begingroup$

                  For any $m$,
                  $$
                  beginalign
                  (1-t)sum_n=0^inftyfract^n1+t^n
                  &=(1-t)sum_n=0^inftysum_k=1^m-1(-1)^k-1t^kn\
                  &+(-1)^m-1(1-t)sum_n=0^inftyfract^mn1+t^n\
                  &=sum_k=1^m-1(-1)^k-1frac1-t1-t^k+O!left(frac1-t1-t^mright)tag1
                  endalign
                  $$
                  Let $tto1^-$,
                  $$
                  lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
                  =sum_k=1^m-1frac(-1)^k-1k+O!left(frac1mright)tag2
                  $$
                  Let $mtoinfty$,
                  $$
                  beginalign
                  lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
                  &=sum_k=1^inftyfrac(-1)^k-1k\[6pt]
                  &=log(2)tag3
                  endalign
                  $$






                  share|cite|improve this answer











                  $endgroup$



                  For any $m$,
                  $$
                  beginalign
                  (1-t)sum_n=0^inftyfract^n1+t^n
                  &=(1-t)sum_n=0^inftysum_k=1^m-1(-1)^k-1t^kn\
                  &+(-1)^m-1(1-t)sum_n=0^inftyfract^mn1+t^n\
                  &=sum_k=1^m-1(-1)^k-1frac1-t1-t^k+O!left(frac1-t1-t^mright)tag1
                  endalign
                  $$
                  Let $tto1^-$,
                  $$
                  lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
                  =sum_k=1^m-1frac(-1)^k-1k+O!left(frac1mright)tag2
                  $$
                  Let $mtoinfty$,
                  $$
                  beginalign
                  lim_tto1^-(1-t)sum_n=0^inftyfract^n1+t^n
                  &=sum_k=1^inftyfrac(-1)^k-1k\[6pt]
                  &=log(2)tag3
                  endalign
                  $$







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited 7 hours ago

























                  answered 13 hours ago









                  robjohn♦robjohn

                  269k27309635




                  269k27309635



























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                      Kathakali Contents Etymology and nomenclature History Repertoire Songs and musical instruments Traditional plays Styles: Sampradayam Training centers and awards Relationship to other dance forms See also Notes References External links Navigation menueThe Illustrated Encyclopedia of Hinduism: A-MSouth Asian Folklore: An EncyclopediaRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to Play10.1353/atj.2005.0004The Illustrated Encyclopedia of Hinduism: A-MEncyclopedia of HinduismKathakali Dance-drama: Where Gods and Demons Come to PlaySonic Liturgy: Ritual and Music in Hindu Tradition"The Mirror of Gesture"Kathakali Dance-drama: Where Gods and Demons Come to Play"Kathakali"Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceMedieval Indian Literature: An AnthologyThe Oxford Companion to Indian TheatreSouth Asian Folklore: An Encyclopedia : Afghanistan, Bangladesh, India, Nepal, Pakistan, Sri LankaThe Rise of Performance Studies: Rethinking Richard Schechner's Broad SpectrumIndian Theatre: Traditions of PerformanceModern Asian Theatre and Performance 1900-2000Critical Theory and PerformanceBetween Theater and AnthropologyKathakali603847011Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceBetween Theater and AnthropologyBetween Theater and AnthropologyNambeesan Smaraka AwardsArchivedThe Cambridge Guide to TheatreRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeThe Garland Encyclopedia of World Music: South Asia : the Indian subcontinentThe Ethos of Noh: Actors and Their Art10.2307/1145740By Means of Performance: Intercultural Studies of Theatre and Ritual10.1017/s204912550000100xReconceiving the Renaissance: A Critical ReaderPerformance TheoryListening to Theatre: The Aural Dimension of Beijing Opera10.2307/1146013Kathakali: The Art of the Non-WorldlyOn KathakaliKathakali, the dance theatreThe Kathakali Complex: Performance & StructureKathakali Dance-Drama: Where Gods and Demons Come to Play10.1093/obo/9780195399318-0071Drama and Ritual of Early Hinduism"In the Shadow of Hollywood Orientalism: Authentic East Indian Dancing"10.1080/08949460490274013Sanskrit Play Production in Ancient IndiaIndian Music: History and StructureBharata, the Nāṭyaśāstra233639306Table of Contents2238067286469807Dance In Indian Painting10.2307/32047833204783Kathakali Dance-Theatre: A Visual Narrative of Sacred Indian MimeIndian Classical Dance: The Renaissance and BeyondKathakali: an indigenous art-form of Keralaeee

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                      Parai AttamNatarajaTandavaRasa lilaLasyaBollywood song and danceHindi dance songsNautchAndhra PradeshAssamHimachal PradeshOdishaPunjabTamil NaduMizoramKerala KalamandalamKerala Lalitakala AcademyKerala Sangeetha Nadaka Academy, ThrissurKerala Folklore AcademyKshetram vadyamKuzhal PattuMappila PaattuPanchari melamPanchavadyamPandi MelamParisha VadyamPulluvan PaattuSopanamThayambakaKerala mural paintingKalaripayattuSarpa KavuTharavadTemplatesPortalCategoryWikiProject KathakaliTheatre in IndiaArts of KeralaClassical theatre of india Malayalamclassical Indian danceMalayalamKeralaCEHindu templesPuranasNatya ShastraNatya ShastraBharataShivaMalayalamKeralaShivaKrishnaKrishnaJayadevaoperaticKrishnaVishnuRamaShivaSuryaYudhishthiraArjunaNalaRavanaDushasanaHiranyakashipuSitaHanumanGarudaHamsaGuṇaSamkhyaHindu philosophysign languagemudrasUnnayi VariyarplaywrightsNalaDamayantimusical modeChristianityMiguel de CervantesJohann Wolfgang von GoetheWilliam ShakespeareGurukulTravancorePalakkadBrahmi...
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