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Analyysi


sarjojenJean le Rond d'Alembertääretöntäsuppenee itseisestisuppenee ehdollisesti





Osamäärätesti tai suhdetesti on tapa tutkia reaali- tai kompleksitermisten sarjojen ∑n=1∞andisplaystyle sum _n=1^infty a_n suppenemista. Testin julkaisi Jean le Rond d'Alembert ja se tunnetaankin joskus nimellä d'Alembertin osamäärätesti. Testiä varten lasketaan sarjan kahden peräkkäisen termin itseisarvon raja-arvo indeksin n lähestyessä ääretöntä ja merkitään saatua raja-arvoa kirjaimella L. Matemaattisesti ilmaistuna


limn→∞|an+1an|=L.=L.

Saatua raja-arvoa tulkitaan seuraavasti:


  • jos L<1displaystyle L<1!, niin sarja suppenee.

  • jos L>1displaystyle L>1!, niin sarja hajaantuu.

  • jos L=1displaystyle L=1!, niin sarjan suppenemisesta ei voida sanoa mitään osamäärätestin perusteella.


Esimerkkejä |



Suppeneva |


Tutkitaan sarjan


∑n=1∞nendisplaystyle sum _n=1^infty frac ne^n

suppenemista. Lasketaan sarjan kahden peräkkäisen termin itseisarvon raja-arvo


limn→∞|an+1an|=limn→∞|n+1en+1nen|=limn→∞|n+1en+1⋅enn|=limn→∞|n+1n⋅enen⋅e|=limn→∞|(1+1n)⋅1e|=1⋅1e=1e<1.=lim _nrightarrow infty left

Koska raja-arvo L=1edisplaystyle L=frac 1e on pienempi kuin 1, niin sarja suppenee.



Hajaantuva |


Tutkitaan sarjan


∑n=1∞enndisplaystyle sum _n=1^infty frac e^nn

suppenemista. Osamäärätestin mukaisesti lasketaan


limn→∞|an+1an|=limn→∞|en+1n+1enn|=limn→∞|en+1n+1⋅nen|=limn→∞|nn+1⋅en⋅een|=limn→∞|(1−1n+1)⋅e|=1⋅e=e>1.=lim _nrightarrow infty left

Koska L=e≈2,718…displaystyle L=eapprox 2,718dots on suurempi kuin 1, niin sarja hajaantuu.



Testi ei kerro suppenemisesta |


Jos sarjan raja-arvo L on tasan 1 eli


limn→∞|an+1an|=1,displaystyle lim _nrightarrow infty left

niin osamäärätestillä ei voida selvittää sen suppenemista.


Esimerkiksi sarja


∑n=1∞1displaystyle sum _n=1^infty 1

hajaantuu, mutta


limn→∞|11|=1.frac 11right

Sarja


∑n=1∞1n2displaystyle sum _n=1^infty frac 1n^2

puolestaan suppenee itseisesti, mutta


limn→∞|1(n+1)21n2|=1.=1.

Sarja


∑n=1∞(−1)n1ndisplaystyle sum _n=1^infty (-1)^nfrac 1n

suppenee ehdollisesti, mutta


limn→∞|(−1)n+1(n+1)(−1)nn|=1.=1.


Katso myös |


  • Vertailuperiaate

  • Juuritesti

  • Integraalitesti


Kirjallisuutta |


  • Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.3, 5.4) ISBN 0-486-60153-6

  • Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 2.36, 2.37) ISBN 0-521-58807-3

  • Pitkäranta, Juhani: Calculus Fennicus. Helsinki: Avoimet oppimateriaalit ry, 2015. ISBN 978-952-7010-12-9, ISBN 978-952-7010-13-6 (pdf).


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