User:CogitoErgoCogitoSum/series

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series $$\sum_{n=1}^\infty a_n$$.

For all tests below, unless otherwise stated:
 * the series we are testing shall be $$\sum_{k=i}^\infty a_k$$.
 * its partial sums shall be $$s_n = \sum_{k=i}^n a_k$$.
 * if it converges, it converges to the value $$S$$.
 * all $$a_k$$ are real-valued.

= Introductory Coursework = In this section the tests and checks taught in high school and junior college coursework are listed.

Basic Checks
These are some of the basic checks, sometimes too trivial or obvious to mention.

Undefined Terms
This may seem trivial or obvious, but it is often overlooked. If any term of the sequence over which the sum is taken is undefined, the sum does not converge. If all terms of the sequence over which the sum is taken is defined, nothing can be said.

It shall be assumed that all sequences are well defined for the remainder of the tests in this article.

Finite Sum
All finite sums - that is, sums over a finite number of sequence terms - must necessarily converge. For an infinite series, however, nothing can be said.


 * If N is a finite value then $$S = \sum_{k=i}^N a_k$$ converges.
 * If N is infinite, nothing can be said.

It shall be assumed that all sequences are infinite - that all sums are infinite series - for the remainder of the tests in this article.

Leading Terms
It is understood that for any series $$\sum_{k=i}^\infty a_k$$, there exists an N such that the series may be separated into $$\big( \sum_{k=i}^N a_k \big) + \big( \sum_{k=N+1}^\infty a_k \big)$$. The left finite sum always converges. Whether or not the overall series converges or diverges depends entirely on the infinite series portion on the right. Thus the finite sum can be ignored, except where calculating the actual value of the series is important. If any test in this article is inapplicable due to the leading terms, the series may be shifted and leading terms ignored arbitrarily, until the conditions of a test are met in the infinite trailing terms. Thus we need only test the series $$\sum_{k=1}^\infty a_{k+N}$$.


 * For any arbitrary N, the series $$\sum_{k=i}^\infty a_k$$ converges if and only if $$\sum_{k=N+1}^\infty a_k $$ converges.

Limit of Partial Sums
If it is possible to explicitly isolate the value of a partial sum algebraically, we may simply test the limit. Whether the series converges or diverges, and to what value it may, depends on $$\lim_{n \to \infty}s_n = S$$.

This is in fact how the infinite series is defined: as the limit of the sequence that is the partial sums.

$$\sum_{k=i}^\infty a_k = \lim_{n\to\infty} \sum_{k=i}^n a_k = \lim_{n\to\infty} s_n = S$$

Not every partial sum can be written algebraically, without summation notation, and in closed-form, as an explicit function of n. But if it can be, and the limit is evaluable, then the sum is determinable and therefore whether or not the series convergence or diverges.

Divergence Test
Sometimes called the Term test, Limit of the summand, nth-term test, test for divergence.

Evaluate the limit of the sequence terms (the summand), $$\lim_{n \to \infty}a_n = L$$.


 * If the limit is undefined or nonzero, that is $$L \ne 0$$, then the series must diverge.
 * If the limit is zero, $$L = 0$$, The test is inconclusive.

Geometric Series Test
The geometric series $$\sum_{k=i}^{\infty} ar^k$$ converges if $$|r| < 1$$, and moreover $$S = a\frac{r^{i}}{1-r}$$.

$p$-Series Test
The series $$\sum_{k=1}^{\infty} \frac{1}{k^p}$$ converges if $$p > 1$$.

The case of $$p = 1$$ yields the harmonic series, which diverges. The case of $$p = 2$$ is the Basel problem and the series converges to $$\frac{\pi^2}{6}$$.

In general, for real number $$p > 1$$, the series is equal to the Riemann zeta function applied to $$p$$, that is $$\zeta(p)$$, and for even p the sum is the p-power of pi times some rational number. For odd p there are no known closed-form expressions.

Alternating Series Test
The alternating series test is also known as the Leibniz criterion.

Suppose the following statements are true:


 * $$ a_n $$ are all positive (or all negative) (i.e. sequence $$\{ (-1)^{k} a_k \}$$ strictly alternate sign),
 * $$ \lim_{n \to \infty} a_n = 0 $$ (i.e. passes the divergence test), and
 * for every n, $$ a_{n+1} \le a_n $$ (monotonic non-increasing magnitudes).

Then $$ \sum_{k = 1}^\infty (-1)^{k} a_k $$ and $$ \sum_{k = 1}^\infty (-1)^{k+1} a_k $$ are convergent series.

Moreover, the error between any partial sum and the true sum is bounded, $$\left|S - s_k \right| < \left| a_{k+1} \right| $$.

Telescoping Series
For some series $$ \sum_{k = i}^\infty a_k $$, let there exist another sequence $$\{b_k\} $$ such that:


 * $$ a_k = b_k - b_{k+1}$$.
 * $$ \lim_{n \to \infty} b_n = L $$.

Then $$ \sum_{k = i}^\infty a_k $$ converges if the limit L exists and is finite.

Moreover, the sum is, $$S = b_i - L $$.

Absolute Convergence Test
Every absolutely convergent series converges. That is,
 * if $$\sum_{k=i}^\infty \left|a_k \right|$$ converges then $$\sum_{k=i}^\infty a_k$$ converges.

The series $$\sum_{k=i}^\infty a_k$$ is said to be absolutely convergent if $$\sum_{k=i}^\infty \left|a_k \right|$$ is also convergent.

If the series $$\sum_{k=i}^\infty a_k$$ is convergent but $$\sum_{k=i}^\infty \left|a_k \right|$$ is not convergent, then the series is said to be conditionally convergent.

d'Alembert's Ratio Test
The ratio test is also known as d'Alembert's criterion, d'Alembert's test, or simply the ratio test.

Evaluate the limit:
 * $$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = r$$

If $$r < 1$$, then the series is absolutely convergent. If $$r > 1$$, including infinite, then the series diverges. If $$r = 1$$, or doesnt exist, the ratio test is inconclusive.

Special Caveats, Ratio
These caveats are not typically taught in the introductory courses, but certain special exceptions exist and are noteworthy.

If the ratio fails to converge due to oscillatory behavior then, we have convergence if:
 * $$\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| < 1$$

And we have divergence if:
 * $$\liminf_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| > 1$$

If the ratio tends to 1, then we also have divergence if the limit is approached strictly from the positive:
 * $$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = 1^+$$

Any other case is inconclusive.

Root Test
The root test is also known as the nth root test (or Cauchy's criterion ??).

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

If $$r < 1$$, then the series converges absolutely. If $$r > 1$$, then the series diverges. If $$r = 1$$, the root test is generally 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 the converse is not necessarily true.

Special Caveats, Roots
Just as with the ratio test, the roots test has comparable caveats.

We can evaluate instead the limit superior, if the limit alone doesnt exist:
 * $$r=\limsup_{n\to\infty}\sqrt[n]{|a_n|}$$

If $$r < 1$$, then the series converges absolutely. If $$r > 1$$, then the series diverges. If $$r = 1$$, the test is inconclusive.

Additionally, whether the limit or the limit superior is used, we may also check that, $$r = 1^+$$, the limit approaches 1 strictly from the positive, thus confirming divergence.

Limit Comparison Test
The Limit comparison test requires the use of a benchmark series $$\sum_{k=i}^\infty b_k$$, whose convergence or divergence is known.

Let the sequences be positive, $$\{a_n\},\{b_n\}>0$$, the limit of the absolute value of the ratios is tested, $$\lim_{n\to\infty} \left|\frac{a_n}{b_n}\right| = L$$.

If L exists, is finite and is some non-zero positive value, $$0 < L < \infty$$, then either both series converge or both series diverge.

Special Caveats, Limit Comparison
Though the limit comparison is generally taught in the introductory courses, these special caveats usually are not.

In the case that $$L \to \infty$$:
 * If the limit of the benchmark sequence converges to a non-zero value or diverges, $$\lim_{n\to\infty} b_n \not\to 0$$, then the series diverges.

In the case that $$L = 0$$:
 * If the limit of the benchmark sequence does not diverge infinitely, $$\lim_{n\to\infty} b_n \not\to \infty$$, then the series converges.

Direct Comparison Test
The direct comparison test can be used to compare the test series to another series, $$\sum_{n=i}^\infty b_n$$, called a benchmark series whose convergence or divergence is known.

Let both sequences be strictly positive, $$a_k,b_k\ge 0$$:
 * If the benchmark series converges and $$|a_k|\le |b_k|$$, then the series converges.
 * If the benchmark series diverges and $$|a_k|\ge |b_k|$$, then the series diverges.
 * Any other case is inconclusive.

Special Caveats, Direct Comparison
The direct comparison test can be generalized such that the sequences do not need to be positive. The first condition on positivity can be modified:

Let both sequences be such that the signs of the associated terms are the same. The following conditions are equivalent:
 * $$\mathrm{sgn}(a_k) = \mathrm{sgn}(b_k)$$
 * $$a_k \cdot b_k\ge 0$$

All of the conditional cases in the above list still hold.

Integral Test
The series can be compared to an integral, using the integral test, to establish convergence or divergence. Let the function $$f:[i,\infty)\to\R^{+}$$ be non-negative and continuous, such that $$f(k) = a_k$$ for all integer k. Evaluate:

$$J = \int_i^\infty f(x) \, dx$$

The series converges if and only if the integral converges:
 * If the integral converges, $$J<\infty$$, then the series converges.
 * If the integral diverges, then the series does so as well.

Moreover, if the function is monotonically non-increasing then the sum S can be bounded, $$J\le S \le J + a_i$$.

= More Advanced Methods = In this section we cover the methods taught in bachelor through doctorate coursework.

Abel's Test
Abel's test allows us to test the series by breaking its terms up into a product of two separate sequences. Let there exist two new sequences $$\{b_k\}, \{c_k\}$$ such that $$a_k = b_k c_k$$.

Suppose the following statements are true about $$\{b_k\}$$:
 * $$\left\{b_k\right\}$$ is a monotonic sequence, and
 * $$\left\{b_k\right\}$$ is bounded.

And suppose the following statements are true about $$\{c_k\}$$:
 * $$\sum_{k=i}^\infty c_n $$ is a convergent series,

Then $$\sum_{k=i}^\infty a_k = \sum_{k=i}^\infty b_k c_k $$ is also convergent.

This test is particularly useful in series that are not absolutely convergent.

This test cannot be used to verify divergence. It is strictly a test for convergence. If any of the conditions are not met, the test is inconclusive.

Dirichlet's Test
Dirichlet's test is similar to Abel's test. The test allows us to test the series by breaking its terms up into a product of two separate sequences. Let there exist two new sequences $$\{b_k\}, \{c_k\}$$ such that $$a_k = b_k c_k$$.

Let $$\{b_k\}$$ be a sequence of real number and $$\{c_n\}$$ be a sequence of complex numbers.

Suppose the following statements are true about $$\{b_k\}$$:
 * $$b_k \geq b_{k+1}$$, monotonic non-increasing.
 * $$\lim_{k \rightarrow \infty}b_k = 0$$.

And suppose the following statements are true about $$\{c_k\}$$:
 * $$\left|\sum^{N}_{k=i}c_k\right|\leq M$$ for every positive integer N (all partial sums of $$\{c_k\}$$ are bounded).

where M is some constant.

If the above conditions are meth then the series $$\sum_{k=i}^\infty a_k = \sum_{k=i}^\infty b_k c_k $$ is also convergent.

This test is particularly useful in series that are not absolutely convergent.

This test cannot be used to verify divergence. It is strictly a test for convergence. If any of the conditions are not met, the test is inconclusive.

Raabe–Duhamel's Ratio Test
Also called Raabe's ratio test, simply Raabe's test, or Raabe–Duhamel's test.

Let $$a_n$$ be a sequence of positive numbers.

Evaluate the limit


 * $$L = \lim_{n\to\infty} n \left( \frac{a_n}{a_{n+1}} - 1 \right)$$

Three possibilities exist:
 * if $$L > 1$$, the series converges (this includes the case $$L \to \infty$$)
 * if $$L < 1$$, the series diverges
 * and if $$L = 1$$ (or if L fails to converge due to oscillation), the test is inconclusive.

This test is an excellent follow-up test to the d'Alembert's ratio test in the case that the traditional ratio test tends to 1, in which case Raabe's test is stronger. Though Raabe's test can still fail due to the limit tending to 1, it is unlikely to fail due to oscillation if the traditional d'Alembert's ratio test is performed first and fails tending to 1.

Alternative Formulation
An alternative formulation of this test is as follows. Let $$\{a_n\}$$ be a sequence of real numbers. Let there exist a natural number $N$. If $$\forall n, n>N$$:


 * $$\left|\frac{a_{n+1}}{a_n}\right|\le 1-\frac{b_n}{n} $$

where $$b_n > 1$$, then the series is convergent.

Cauchy Condensation Test
The Cauchy condensation test allows us to subsample the sequence over which the sum is evaluated.

Let $$\left \{ a_n \right \}$$ be a non-negative non-increasing sequence. Then the sum $$S = \sum_{n=1}^\infty a_n$$ converges if and only if the sum $$A = \sum_{n=0}^\infty 2^n a_{2^n}$$ converges.

Moreover, if they converge, then the sum can be bounded, $$\frac{1}{2} A \leq S \leq A$$.

Generalization, Cauchy Condensation
The subsampling can be generalized with alternative base values, b.

The sum $$S = \sum_{n=1}^\infty a_n$$ converges if and only if the sum $$A = \sum_{n=0}^\infty b^n a_{b^n}$$ converges.

If they converge, then the sum is bounded thus, $$\frac{1}{b} A \leq S \leq A$$.

Cauchy's Convergence Criterion
Also called Cauchy's convergence test. This is not a practical test for series. It is used mostly for theoretical purposes, and proving tests that are far more practical.

A series $$\sum_{k=i}^\infty a_k$$ is convergent if and only if for every $$\varepsilon>0$$ there is a natural number $N$ such that, $$\forall n, n > N$$ and $$\forall p, p>1$$, the following statement holds.


 * $$|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon$$

Frink's Ratio Test
Also called the exponential test, this test is a good follow-up when the d'Alembert Ratio test fails due to convergence to 1. This test is stronger, and occasionally more convenient to implement.

Evaluate the limits:
 * $$L = \limsup_{n\to\infty}\left(\frac{a_{n+1}}{a_n}\right)^n$$
 * $$l = \liminf_{n\to\infty}\left(\frac{a_{n+1}}{a_n}\right)^n$$

Then we have:
 * Convergence if $$L<\frac1e$$.
 * Divergence if $$l>\frac1e$$.
 * The test is inconclusive otherwise.

Weierstrass M-test
Suppose that (fn) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (Mn) satisfying the conditions Then the series
 * $$|f_n(x)|\leq M_n$$ for all $$n \geq 1$$ and all $$x \in A$$, and
 * $$\sum_{n=1}^{\infty} M_n $$ converges.
 * $$\sum_{n=1}^{\infty} f_n (x)$$

converges absolutely and uniformly on A.

Extensions to the ratio test
The ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case.

Bertrand's test
Bertrand's test is useful when the d'Alembert's Ratio test fails due to approaching 1.

Let { an } be a sequence of positive numbers.

Define and evaluate


 * $$L=\lim_{n\to\infty} \ln(n)\left(n\left(\frac{a_n}{a_{n+1}}-1 \right)-1\right).$$

If L exists, there are three possibilities:


 * if L > 1 the series converges (this includes the case L = ∞)
 * if L < 1 the series diverges
 * and if L = 1 the test is inconclusive.

Gauss's test
Let { an } be a sequence of positive numbers. If $$\frac{a_n}{a_{n + 1}} = 1+ \frac{\alpha}{n} + O(1/n^\beta)$$ for some β > 1, then $$ \sum a_n$$ converges if $α > 1$ and diverges if $α ≤ 1$.

Kummer's test
Let { an } be a sequence of positive numbers. Then:

(1) $$ \sum a_n$$ converges if and only if there is a sequence $$b_{n}$$ of positive numbers and a real number c > 0 such that $$b_k (a_{k}/a_{k+1}) - b_{k+1} \ge c$$.

(2) $$ \sum a_n$$ diverges if and only if there is a sequence $$b_{n}$$ of positive numbers such that $$b_k (a_{k}/a_{k+1}) - b_{k+1} \le 0$$

and $$ \sum 1/b_{n}$$ diverges.