Ratio test. Root test or n th root test. Suppose that the terms of the sequence in question are non-negative.
Define r as follows:. 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. Integral test. The series can be compared to an integral to establish convergence or divergence. Limit comparison test. Alternating series test. Cauchy condensation test. An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long.
The power series of the exponential function is absolutely convergent everywhere. The path formed by connecting the partial sums of a conditionally convergent series is infinitely long. The power series of the logarithm is conditionally convergent.
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The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges. 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. From Wikipedia, the free encyclopedia. All backorders will be released at the final established price.
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