In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.
Suppose that we have two series and with for all .
Then if with , then either both series converge or both series diverge.
Because we know that for every there is a positive integer such that for all we have that , or equivalently
As we can choose to be sufficiently small such that is positive.
So and by the direct comparison test, if converges then so does .
Similarly , so if diverges, again by the direct comparison test, so does .
That is, both series converge or both series diverge.
We want to determine if the series converges. For this we compare it with the convergent series
As we have that the original series also converges.
One can state a one-sided comparison test by using limit superior. Let for all . Then if with and converges, necessarily converges.
Let and for all natural numbers . Now
does not exist, so we cannot apply the standard comparison test. However,
and since converges, the one-sided comparison test implies that converges.
Let for all . If diverges and converges, then necessarily
that is,
The essential content here is that in some sense the numbers are larger than the numbers .
Let be analytic in the unit disc and have image of finite area. By Parseval's formula the area of the image of is proportional to . Moreover,
diverges.
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In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Suppose that we have two series and with for all . Then if with , then either both series converge or both series diverge. Because we know that for every there is a positive integer such that for all we have that , or equivalently As we can choose to be sufficiently small such that is positive. So and by the direct comparison test, if converges then so does .
In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. Consider an integer N and a function f defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series converges to a real number if and only if the improper integral is finite. In particular, if the integral diverges, then the series diverges as well.
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence 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 the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number.