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Series – Types & Tests
series is an infinite sum, where the n th summand is the n th term of a sequence. A series is usually denoted using “sigma notation,” i.e.,
The index n may begin with 0, 1, or k for any natural number k, as a matter of convenience.
Types of Series
HARMONIC SERIES
This series does not converge, because its sequence of partial sums is unbounded. To see this, notice that we have the following:
GEOMETRIC SERIES
where a and r are both constants, and r is called the common ratio. This series clearly diverges if r is greater than or equal to 1. If r is between 0 and 1, then the series converges and the sum is given by:
P-SERIES
where the common exponent p is a positive real constant. This series clearly diverges if p is less than or equal to 1, by comparison with the harmonic series. It converges if p is strictly greater than 1, though what the sum is in that case is known in only a few cases. It is notable that for p = 2 (the sum of the inverses of the squares), the sum is p2/6 (pi squared over six).
ALTERNATING SERIES
Any series in which the terms alternate in sign, for example:
Any such series converges if and only if its terms decrease (in absolute value), with zero as a limit.
POWER SERIES
where each ai is a distinct constant. It can be shown that any such series either converges at x = 0, or for all real x, or for all x with –R < x < R for some positive real R. The interval (–R, R) is called the radius of convergence. This important series should be thought of as a function in x for all x in the radius of convergence. Where defined, this function has derivatives of all orders, and may be differentiated and integrated “term-by-term.” That is,
Convergence Tests
A series converges when its sequence of partial sums converges, that is, if the sequence of values given by the first term, then the sum of the first two terms, then the sum of the first three terms, etc., converges as a sequence. A series is said to converge absolutely if the series still converges when all of the terms of the series are made non-negative (by taking their absolute value). A series which converges but does not converge absolutely is said to converge conditionally (cf. alternating series above). A series which does not converge is said to diverge. All of the tests below except for the nth-term test are for series with non-negative terms only.
N th-TERM TEST
In any series, if the terms of the series do not diminish in absolute value to zero (i.e., if the limit of the sequence of terms is not zero), then the series diverges. This important test must be used with care: it is not a test for convergence, and must not be interpreted to mean that any series whose sequence of terms does limit to zero necessarily converges. The harmonic series (above) is a good counter example.
COMPARISON TEST
If
are any two series, all of whose terms are non-negative, such that an < bn for all but finitely many n, then if the first series diverges the second one does also. Conversely, if the second series converges the first one does also.
LIMIT COMPARISON TEST
If
are any two series, all of whose terms are non-negative, and if the limit
exists and is a finite real number greater than zero, then either both series converge or both series diverge.
INTEGRAL TEST
If the terms of a series are positive and decreasing, then the series and the integral
either both converge or both diverge.
RATIO TEST
If the terms of a given series are all non-negative, and if the limit
exists, then the series converges if L is less than 1, and diverges if L is greater than 1. If L is equal to 1, the test is inconclusive.
ROOT TEST
If the terms of a given series are all non-negative, and if the limit
exists, then the series converges if L is less than 1, and diverges if L is greater than 1. If L is equal to 1, the test is inconclusive.
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