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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.

مثال عملي · does n/2^n for n = 1 to oo converge

Calculus · Series-And-Sums

Convergence of n/2^n

2^{- n} n

خطوة خطوة

  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.

كشفت الإجابة
\text{converges to } 2

الآن أنت اختار مشكلة، أو طبعها أو رسمها بنفسك، كل خطوة، صورة، الإجابة مخفية حتى تسأل.

إبقي عملك الخاص

إن الحساب المجاني يضيفه ملاحظات على كل درس، وسجل لما أنجزته، ومشاكلك التي حلتها في مكان واحد، ومعلم يمكنك أن تسأله عن هذه الصفحة. والرياضيات نفسها مفتوحة أمام الجميع، سواء كانوا مستخدمين أو غير مستخدمين.

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لمس أي رمز للحصول على التعريف الكامل، والصورة، وما يعنيه كل حرف في ذلك.

كيف: 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.

أسئلة يطرحها الناس

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.

أكثر في 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