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The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers.
This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit. However, if the original series diverges, then the grouped series do not necessarily diverge, as in this example of Grandi's series above.
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. Statement [ edit ]
(Monotone convergence theorem) If is bounded and monotonic for all greater than some , then it is convergent. A sequence is convergent if and only if every subsequence is convergent. If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point.
The more general class of p-series, =, exemplifies the possible results of the test: If p ≤ 0, then the nth-term test identifies the series as divergent. If 0 < p ≤ 1, then the nth-term test is inconclusive, but the series is divergent by the integral test for convergence.
Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally convergent series.