Definitions

Convergence_tests

Convergence tests

In mathematics, Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series.

List of Tests

  • Ratio test. Assume that for all n, a_n > 0. Suppose that there exists r such that

lim_{n to infty} left|frac{a_{n+1}}{a_n}right| = r.
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

  • Root test or nth root test. Define r as follows:

r = limsup_{nrightarrowinfty}sqrt[n]
>,

where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).

If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

int_{1}^{infty} f(x), dx = lim_{t to infty} int_{1}^{t} f(x), dx < infty,
then the series converges. But if the integral diverges, then the series does so as well.

  • Limit comparison test. If left { a_n right }, left { b_n right } > 0, and the limit lim_{n to infty} frac{a_n}{b_n} exists and is not zero, then sum_{n=1}^infty a_n converges if and only if sum_{n=1}^infty b_n converges.

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Comparison

The root test is stronger than the ratio test (it is more powerful because the required condition is weaker): whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.

For example, for the series

1 + 1 + 0.5 + 0.5 + 0.25 + 0.25 + 0.125 + 0.125 + ...

convergence follows from the root test but not from the ratio test.

Examples

Consider the series

(*) ;;; sum_{n=1}^{infty} frac{1}{n^alpha}.

Cauchy condensation test implies that (*) finitely convergent if

(**) ;;; sum_{n=1}^{infty} 2^n left (frac{1}{2^n}right )^alpha

finitely convergent. Since

sum_{n=1}^{infty} 2^n left (frac{1}{2^n}right )^alpha = sum_{n=1}^{infty} 2^{n-nalpha} = sum_{n=1}^{infty} 2^{(1-alpha)^n}

(**) is geometric series with ratio 2^{(1-alpha)} . (**) is finitely convergent if its ratio is less than one (namely alpha > 1). Thus, (*) is finitely convergent if and only if alpha > 1 .

The Tests: When to use & Examples

http://www.math.cornell.edu/~alozano/calculus/testconvergence.pdf

References

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