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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.

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

Calculus · Series-And-Sums

Convergence of n/2^n

2^{- n} n

Àwọn ìṣàmúlò-ètò

\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.

Fi àwọn àgbèwọlé hàn

\text{converges to } 2

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Bii o ṣe le: 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.

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 Series and sums 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