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 \(\sum_{n=1}^\infty a_n\).

If the limit of the summand is undefined or nonzero, that is \(\lim_{n \to \infty}a_n \ne 0\), then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.

This is also known as d'Alembert's criterion.

Consider two limits \(\ell=\liminf_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\) and \(L=\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\). If \(\ell>1\), the series diverges. If \(L<1\) then the series converges absolutely. If \(\ell\le1\le L\) then the test is inconclusive, and the series may converge absolutely or conditionally or diverge.

This is also known as the nth root test or Cauchy's criterion.

Let

\(r=\limsup_{n\to\infty}\sqrt[n]{|a_n|},\)

where \(\limsup\) denotes the limit superior (possibly \(\infty\); if the limit exists it is the same value).

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

The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.

The series can be compared to an integral to establish convergence or divergence. Let \(f:[1,\infty)\to\R_+\) be a non-negative and monotonically decreasing function such that \(f(n) = a_n\). 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. In other words, the series \({a_n}\) converges if and only if the integral converges.

If the series \(\sum_{n=1}^\infty b_n\) is an absolutely convergent series and \(|a_n|\le |b_n|\) for sufficiently large n , then the series \(\sum_{n=1}^\infty a_n\) converges absolutely.

If \(\{a_n\},\{b_n\}>0\), (that is, each element of the two sequences is positive) and the limit \(\lim_{n\to\infty} \frac{a_n}{b_n}\) exists, is finite and non-zero, then either both series converge or both series diverge.

Let \(\left \{ a_n \right \}\) be a non-negative non-increasing sequence. Then the sum \(A = \sum_{n=1}^\infty a_n\) converges if and only if the sum \(A^* = \sum_{n=0}^\infty 2^n a_{2^n}\) converges. Moreover, if they converge, then \(A \leq A^* \leq 2A\) holds.

Suppose the following statements are true:

1. \(\sum a_n\) is a convergent series,
2. \(\left\{b_n\right\}\) is a monotonic sequence, and
3. \(\left\{b_n\right\}\) is bounded.

Then \(\sum a_nb_n\) is also convergent.

Suppose the following statements are true:

• \((a_n)_{n=1}^\infty\) is monotonic,
• \(\lim_{n \to \infty} a_n = 0\)

Then \(\sum_{n = 1}^\infty (-1)^{n} a_n\) and \(\sum_{n = 1}^\infty (-1)^{n+1} a_n\) are convergent series. This test is also known as the Leibniz criterion.

If \(\{a_n\}\) is a sequence of real numbers and \(\{b_n\}\) a sequence of complex numbers satisfying

• \(a_n \geq a_{n+1}\)

• \(\lim_{n \rightarrow \infty}a_n = 0\)

• \(\left|\sum^{N}_{n=1}b_n\right|\leq M\) for every positive integer N

where M is some constant, then the series

\(\sum^{\infty}_{n=1}a_n b_n\)

converges.

A series \(\sum_{i=0}^\infty a_i\) is convergent if and only if for every \(\varepsilon>0\) there is a natural number N such that

\(|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon\)

holds for all n > N and all p ≥ 1.

Let \((a_n)_{n \geq 1}\) and \((b_n)_{n \geq 1}\) be two sequences of real numbers. Assume that \((b_n)_{n \geq 1}\) is a strictly monotone and divergent sequence and the following limit exists:

\(\lim_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}=l.\)

Then, the limit

\(\lim_{n \to \infty} \frac{a_n}{b_n}=l.\)

Suppose that (f_n) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (M_n) satisfying the conditions

• \(|f_n(x)|\leq M_n\) for all \(n \geq 1\) and all \(x \in A\), and
• \(\sum_{n=1}^{\infty} M_n\) converges.

Then the series

\(\sum_{n=1}^{\infty} f_n (x)\)

converges absolutely and uniformly on A.