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Infinite series and convergence tests

Geometric, p-series, ratio and comparison tests, alternating series.

A series converges if its partial sums settle. Geometric series converge for |r| < 1; p-series for p > 1; the ratio test decides most series with factorials or powers. Picture it: the partial sums as a sequence creeping toward a ceiling. Think it: convergence is a statement about tails: what happens after any finite start is irrelevant.

ឧទាហរណ៍​ដែល​បាន​ធ្វើ · does n/2^n for n = 1 to oo converge

Calculus · Series-And-Sums

Convergence of n/2^n

2^{- n} n

ជំហាន​ដោយ​ជំហាន

  1. \sum_{n=1}^{\infty} 2^{- n} n

    Does this series converge?

  2. \lim_{n\to\infty} 2^{- n} n = 0

    Divergence test: if the terms do not go to 0 the series cannot converge.

  3. \lim \left|\frac{a_{k+1}}{a_k}\right| = \lim \frac{\left|{\frac{n + 1}{n}}\right|}{2} = \frac{1}{2}

    Ratio test: < 1 converges, > 1 diverges, = 1 says nothing.

  4. = 2

    It converges, and the CAS knows the sum.

បង្ហាញ​ចម្លើយ
\text{converges to } 2

ឥឡូវ​អ្នក ជ្រើស​បញ្ហា ឬ​វាយ​បញ្ចូល ឬ​គូរ​របស់​អ្នក​ផ្ទាល់ ។ ជំហាន​នីមួយៗ រូបភាព ចម្លើយ​ត្រូវ​បាន​លាក់​រហូត​ដល់​អ្នក​សួរ ។

រក្សា​ការ​ធ្វើការ​របស់​អ្នក​ផ្ទាល់

គណនី​ឥត​គិត​ថ្លៃ​បន្ថែម​ចំណាំ​លើ​មេរៀន​នីមួយៗ កំណត់​ហេតុ​នៃ​អ្វី​ដែល​អ្នក​បាន​បញ្ចប់ បញ្ហា​ដែល​អ្នក​បាន​ដោះស្រាយ​នៅ​កន្លែង​មួយ និង​គ្រូ​ដែល​អ្នក​អាច​សួរ​អំពី​ទំព័រ​នេះ ។ គណិតវិទ្យា​ដោយ​ខ្លួន​វា​គឺ​បើក​ចំហ​សម្រាប់​អ្នក​ទាំងអស់​គ្នា ចុះឈ្មោះ​ចូល ឬ​មិន​ចុះឈ្មោះ​ចូល ។

ចុះឈ្មោះ ចូល

និមិត្ត​សញ្ញា​ដែល​បាន​ប្រើ​នៅ​ទីនេះ

ចុច​និមិត្ត​សញ្ញា​ណាមួយ​សម្រាប់​ការ​កំណត់​ពេញលេញ រូបភាព និង​តួអក្សរ​នីមួយៗ​មាន​ន័យ​អ្វី ។

របៀប​: Infinite series and convergence tests

  1. Does this series converge?
  2. Divergence test: if the terms do not go to 0 the series cannot converge.
  3. Ratio test: < 1 converges, > 1 diverges, = 1 says nothing.
  4. It converges, and the CAS knows the sum.

សំណួរ​ដែល​មនុស្ស​សួរ

What is a derivative in one sentence?

The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.

បន្ថែម​ទៀត​ក្នុង Calculus

LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc length