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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.

工作范例 · does n/2^n for n = 1 to oo converge

Calculus · Series-And-Sums

Convergence of n/2^n

2^{- n} n

一步

\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.

发送答案

\text{converges to } 2

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如何: 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.

人们对人们提出的问题

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.