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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.
Ví dụ đã làm · does n/2^n for n = 1 to oo converge
Bước một bước
- \sum_{n=1}^{\infty} 2^{- n} n
Does this series converge?
- \lim_{n\to\infty} 2^{- n} n = 0
Divergence test: if the terms do not go to 0 the series cannot converge.
- \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.
- = 2
It converges, and the CAS knows the sum.
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Làm thế nào: Infinite series and convergence tests
- Does this series converge?
- Divergence test: if the terms do not go to 0 the series cannot converge.
- Ratio test: < 1 converges, > 1 diverges, = 1 says nothing.
- It converges, and the CAS knows the sum.
Câu hỏi người ta hỏi
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.
More in 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