A sum with finitely many terms has a closed form found by pattern or induction; an infinite series converges when its partial sums approach a limit. Σ 1/n² = π²/6 is the Basel problem, Euler's first famous result.
Rechenbeispiel · sum of 1/n^2 for n = 1 to oo
Schritt für Schritt
- \sum_{n=1}^{\infty} \frac{1}{n^{2}}
Write the sum out.
- 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots
The first few terms.
- S_{3} \approx 1.3611, S_{6} \approx 1.4914, S_{11} \approx 1.5580, S_{51} \approx 1.6255
Partial sums approach the limit.
- = \frac{\pi^{2}}{6} \approx 1.6449
Infinite series: this converges, and the closed form is the limit of the partial sums.
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Wie man: Series and sums
- Write the sum out.
- The first few terms.
- Partial sums approach the limit.
- Infinite series: this converges, and the closed form is the limit of the partial sums.
Fragen, die die Leute stellen
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.
Mehr in Calculus
LimitsDerivativesIntegralsDefinite integralsTaylor seriesMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests