In mathematics, a series is the sum of the terms of a sequence of numbers.
Given a sequence , the nth partial sum is the sum of the first n terms of the sequence, that is,
Examples of convergent and divergent series
- The reciprocals of the positive integers produce a divergent series (harmonic series):
- Alternating the signs of the reciprocals of positive integers produces a convergent series:
- Alternating the signs of the reciprocals of positive odd integers produces a convergent series (the Leibniz formula for pi):
- The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"):
- The reciprocals of triangular numbers produce a convergent series:
- The reciprocals of factorials produce a convergent series (see e):
- The reciprocals of square numbers produce a convergent series (the Basel problem):
- The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"):
- Alternating the signs of reciprocals of powers of 2 also produces a convergent series:
- The reciprocals of Fibonacci numbers produce a convergent series:
Convergence testsThere are a number of methods of determining whether a series converges or diverges.
for all n, , and converges, then so does
for all n, , and diverges, then so does
Ratio test. Assume that for all n, . Suppose that there exists such that
Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
- where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).
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. Let be a positive and monotone decreasing function. If
Limit comparison test. If , and the limit exists and is not zero, then converges if and only if converges.
Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form , if is monotone decreasing, and has a limit of 0 at infinity, then the series converges.
Cauchy condensation test. If is a positive monotone decreasing sequence, then converges if and only if converges.
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