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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. Any series that is not convergent is said to be divergent or to diverge.
The test works because the space of real numbers and the space of complex numbers (with the metric given by the absolute value) are both complete. From here, the series is convergent if and only if the partial sums:= = are a Cauchy sequence.
An alternative formulation of this test is as follows. Let { a n} be a series of real numbers. Then if b > 1 and K (a natural number) exist such that | + | for all n > K then the series {a n} is convergent.
Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the Riemann series theorem. [31] [32] [33]
Notably, these series provide examples of infinite sums that converge or diverge arbitrarily slowly. For instance, in the case of k = 2 {\displaystyle k=2} and α = 1 {\displaystyle \alpha =1} , the partial sum exceeds 10 only after 10 10 100 {\displaystyle 10^{10^{100}}} (a googolplex ) terms; yet the series diverges nevertheless.
The nth partial sum is given by a simple formula: = = (+). This equation was known to the Pythagoreans as early as the sixth century BCE. [5] Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.
The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the ...
There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters .