Series and sums

Partial sums, closed forms and convergence.

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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · sum of 1/n^2 for n = 1 to oo

Calculus · Series-And-Sums

Sum n^(-2) for n = 1 to oo

\sum_{n=1}^{\infty} \frac{1}{n^{2}}

Àwọn ìṣàmúlò-ètò

\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.

Fi àwọn àgbèwọlé hàn

\frac{\pi^{2}}{6}

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Bii o ṣe le: 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.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Calculus

Limits Derivatives Integrals Definite integrals Taylor series Maxima and minima The chain rule Implicit differentiation Related rates and optimisation Integration techniques: substitution, parts, partial fractions Applications of integration: area, volume, arc length Infinite series and convergence tests