Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

defines a series S that is denoted The nth partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if and only if the sequence

of its partial sums tends to a limit; that means that, when adding one

More precisely, a series converges, if and only if there exists a number

such that for every arbitrarily small positive number

, If the series is convergent, the (necessarily unique) number

is called the sum of the series.

The same notation is used for the series, and, if it is convergent, to its sum.

This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b.

There are a number of methods of determining whether a series converges or diverges.

such that If r < 1, then the series is absolutely convergent.

If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

Suppose that the terms of the sequence in question are non-negative.

Define r as follows: If r < 1, then the series converges.

If r = 1, the root test is inconclusive, and the series may converge or diverge.

The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations.

In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true.

The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series.

The series can be compared to an integral to establish convergence or divergence.

be a positive and monotonically decreasing function.

But if the integral diverges, then the series does so as well.

Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form

is monotonically decreasing, and has a limit of 0 at infinity, then the series converges.

is a positive monotone decreasing sequence, then

Every absolute convergent series (real or complex) is also convergent, but the converse is not true.

The Maclaurin series of the exponential function is absolutely convergent for every complex value of the variable.

The Maclaurin series of the logarithm function

Agnew's theorem characterizes rearrangements that preserve convergence for all series.

of partial sums defined by converges uniformly to f. There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test.

The Cauchy convergence criterion states that a series converges if and only if the sequence of partial sums is a Cauchy sequence.