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.
Намунаи коркардашуда · sum of 1/n^2 for n = 1 to oo
Қадами ба қадам
- \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.
Ҷавоби ҷавобро нишон диҳед
Ҳоло шумо Масъаларо интихоб кунед, ё ки онро навишта ё кашед. Ҳар қадам, тасвир, ҷавоб то он даме, ки шумо пурсед, пинҳон мешавад.
Дар ҳисоби ройгон ба ҳар як дарс қайдҳо илова карда мешаванд, қайди он, ки шумо чӣ кор кардаед, масъалаҳои ҳалшуда дар як ҷо ва муаллиме, ки шумо метавонед дар бораи ин саҳифа пурсед. Математика барои ҳама кушода аст, хоҳ ворид шуда бошад ё на.
Бақайдгирӣ Ворид шуданСимволы, используемые здесь
Барои муайянкунии пурраи маъно, тасвир ва маънои ҳар як ҳарф дар он, ба ҳар як аломат пахш кунед.
Чӣ тавр: 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.
Саволҳои маъмул
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.
& Иловаи забон 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