Ad
related to: sum of convergent numbers example problems worksheet answers 3rd level
Search results
Results From The WOW.Com Content Network
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.
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.
For example, the sum of 1/n where n has at most one 9, is a convergent series. But the sum of 1/n where n has no 9 is convergent. Therefore, the sum of 1/n where n has exactly one 9, is also convergent. Baillie [11] showed that the sum of this last series is about 23.04428 70807 47848 31968.
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original ...
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]
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.
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.
For instance, for Alcuin's version of the problem, =: a camel can carry 30 measures of grain and can travel one leuca while eating a single measure, where a leuca is a unit of distance roughly equal to 2.3 kilometres (1.4 mi). The problem has =: there are 90 measures of grain, enough to supply three trips.