Convergence tests

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In mathematics, convergence tests are methods to determine if an infinite series converges or diverges.

If the blue series,

Σ b n

, can be proven to converge, then the smaller series,

Σ a n

must converge. By contraposition, if the red series,

Σ a n

is proven to diverge, then

Σ b n

must also diverge.

Test for divergence. If $\lim_{n \to \infty} a_n \neq 0$ , then $\sum_{n=1}^\infty a_n$ diverges.

Comparison test. The terms of the sequence $\left \{ a_n \right \}$ are compared to those of another sequence $\left \{ b_n \right \}$ . If, for all n,

$0 \le \ a_n \le \ b_n$ , and $\sum_{n=1}^\infty b_n$ converges, then so does $\sum_{n=1}^\infty a_n$ .

However, if, for all n,

$0 \le \ b_n \le \ a_n$ , and $\sum_{n=1}^\infty b_n$ diverges, then so does $\sum_{n=1}^\infty a_n$ .

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. Suppose that the terms of the sequence in question are non-negative, and that there exists r such that

$\lim_{n \to \infty} \sqrt[n]{a_n} = r$

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.

Root test is equivalent to ratio test.

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

$\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty,$

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.

Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form $\sum_{n=1}^\infty a_n (-1)^n$ , if $\left \{ a_n \right \}$ is monotone decreasing, and has a limit of 0, then the series converges.