Definitions

# 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

$lim_\left\{n to infty\right\} left|frac\left\{a_\left\{n+1\right\}\right\}\left\{a_n\right\}right| = r$.
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:

$r = limsup_\left\{nrightarrowinfty\right\}sqrt\left[n\right]$
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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.

• Integral test. The series can be compared to an integral to establish convergence or divergence. Let $f\left(n\right) = a_n$ be a positive and monotone decreasing function. If

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

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

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

Cauchy condensation test implies that (*) finitely convergent if

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

finitely convergent. Since

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

(**) is geometric series with ratio $2^\left\{\left(1-alpha\right)\right\}$. (**) 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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