The idea behind the limit comparison test is that if you take a known convergent series and multiply each of its terms by some number, then that new series also converges. And it doesn’t matter whether the multiplier is, say, 100, or 10,000, or 1/10,000 because any number, big or small, times the finite sum of the original series is still a finite number. The same thing goes for a divergent series multiplied by any number. That new series also diverges because any number, big or small, times infinity is still infinity. This is over simplified — it’s only in the limit that one series is sort of a multiple of the other — but it conveys the basic principle.
You can discover whether such a connection exists between two series by looking at the ratio of the nth terms of the two series as n approaches infinity. Here’s the test.
Limit Comparison Test: For two series,
where L is finite and positive, either both series converge or both diverge.
This is a good test to use when you can’t use the direct comparison test for your series because it goes the wrong way — in other words, your series is bigger than a known convergent series or smaller than a known divergent series.
Here’s an example: Does
converge or diverge? This series resembles the convergent p-series
so that’s your benchmark. But you can’t use the direct comparison test because the terms of your series are bigger than
Instead, you use the limit comparison test.
Take the limit of the ratio of the nth terms of the two series. It doesn’t matter which series you put in the numerator and which in the denominator, but if you put the known, benchmark series in the denominator, this makes it a little easier to do these problems and to grasp the results.
Because the limit is finite and positive, and because the benchmark series converges, your series must also converge.
Let’s try another example. Determine the convergence or divergence of
The limit comparison test is a good one for series, like this one, in which the general term is a rational function — in other words, where the general term is a quotient of two polynomials.
Determine the benchmark series.
Take the highest power of n in the numerator and the denominator — ignoring any coefficients and all other terms — then simplify. Like this:
The benchmark series is thus
the divergent harmonic series.
Take the limit of the ratio of the nth terms of the two series.
Because the limit from Step 2 is finite and positive, and because the benchmark series diverges, your series must also diverge.
Contrary to the formal definition of the Limit Comparison Test (at the beginning of this article), the limit, L, doesn’t have to be finite and positive for the test to work. First, if the benchmark series is convergent, and you put it in the denominator of the limit, and the limit is zero, then your series must also converge. If the limit is infinity, you can’t conclude anything. And second, if the benchmark series is divergent, and you put it in the denominator, and the limit is infinity, then your series must also diverge. If the limit is zero, you learn nothing.
Lengo la mtihani wa kulinganisha kikomo ni kwamba ikiwa unachukua mfululizo unaojulikana na kuzidisha kila moja ya maneno yake kwa idadi fulani, basi mfululizo huo mpya pia unalingana. Na haijalishi kama multiplier ni, sema, 100, au 10,000, au 1 × 10,000, kwa sababu idadi yoyote, kubwa au ndogo, mara ya jumla ya mwisho ya mfululizo wa awali bado ni idadi ya mwisho. Jambo moja linatumika kwa mfululizo divergent kuzidishwa na idadi yoyote. Mfululizo huo mpya pia unatofautiana kwa sababu idadi yoyote, kubwa au ndogo, mara infinity bado ni infinity. Hii ni juu ya kurahisishwa <unk> ni tu katika kikomo kwamba mfululizo mmoja ni aina ya mara nyingi ya nyingine <unk> lakini inatoa kanuni ya msingi. Unaweza kugundua kama uhusiano kama huo upo kati ya mfululizo mbili kwa kuangalia uwiano wa maneno ya n ya mfululizo huo wawili wakati n inakaribia kutokuwa na mwisho. Hapa ni mtihani. Mtihani wa Kulinganisha Mipaka: Kwa mfululizo mbili, ambapo L ni mwisho na chanya, ama mfululizo wote wawili converge au wote wawili kutofautiana. Hii ni mtihani mzuri kutumia wakati huwezi kutumia mtihani wa kulinganisha moja kwa moja kwa mfululizo wako kwa sababu huenda njia mbaya - kwa maneno mengine, mfululizo wako ni kubwa kuliko mfululizo unaojulikana convergent au ndogo kuliko mfululizo unaojulikana divergent. Hapa ni mfano: Je, converge au kutofautiana? Mfululizo huu unafanana na mfululizo wa p-mfululizo hivyo hiyo ni benchmark yako. Lakini huwezi kutumia mtihani wa kulinganisha moja kwa moja kwa sababu maneno ya mfululizo wako ni kubwa kuliko Badala yake, wewe kutumia mtihani wa kulinganisha kikomo. Kuchukua kikomo cha uwiano wa n-th maneno ya mfululizo mbili. Haijalishi ni mfululizo gani unaweka katika mtoaji na ambayo katika mtoaji, lakini ikiwa unaweka mfululizo unaojulikana, wa msingi katika mtoaji, hii inafanya iwe rahisi kufanya matatizo haya na kuelewa matokeo. Kwa sababu kikomo ni mwisho na chanya, na kwa sababu mfululizo benchmark converges, mfululizo wako lazima pia converge. Hebu jaribu mfano mwingine. "Kuchunguza Convergence au Divergence ya ""Limit Comparison Test"" ni moja nzuri kwa mfululizo, kama hii moja, ambapo neno la jumla ni kazi ya busara, yaani, ambapo neno la jumla ni quotient ya polynomials mbili." Kuamua mfululizo benchmark. Kuchukua nguvu ya juu ya n katika numerator na denominator <unk> kupuuza coefficients yoyote na maneno mengine yote <unk> kisha kurahisisha. Kama hii: mfululizo benchmark ni hivyo mfululizo divergent harmonic. Kuchukua kikomo cha uwiano wa n-th maneno ya mfululizo mbili. Kwa sababu kiwango cha hatua ya pili ni cha mwisho na chanya, na kwa sababu mfululizo wa benchmark unatofautiana, mfululizo wako lazima pia utatofautiane. Kinyume na ufafanuzi rasmi wa Mtihani wa Kulinganisha Mipaka (mwanzoni mwa makala hii), kikomo, L, haipaswi kuwa na mwisho na chanya kwa mtihani kufanya kazi. Kwanza, kama mfululizo benchmark ni convergent, na wewe kuweka katika denominator ya kikomo, na kikomo ni sifuri, basi mfululizo wako lazima pia converge. Kama kikomo ni infinity, huwezi kuhitimisha chochote. Kwa mfano, mfululizo wa mlinganisho wa divergent, na mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho wa mlinganisho. Ikiwa kikomo ni sifuri, hujifunzi chochote.