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 and only 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. More precisely, a series converges, if and only if there exists a number such that for every arbitrarily small positive number , there is a (sufficiently large) integer such that for all ,
If the series is convergent, the (necessarily unique) number is called the sum of the series.
The same notation
is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b.
Any series that is not convergent is said to be divergent or to diverge.
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Transcription
Let's say I've got a sequence. It starts at 1, then let's say it goes to negative 1/2. Then it goes to positive 1/3. Then it goes to negative 1/4. Then it goes to positive 1/5. And it just keeps going on and on and on like this. And we could graph it. Let me draw our vertical axis. So I'll graph this as our y-axis. And I'm going to graph y is equal to a sub n. And let's make this our horizontal axis where I'm going to plot our n's. So this right over here is our n's. And this is, let's say this right over here is positive 1. This right over here is negative 1. This would be negative 1/2. This would be positive 1/2. And I'm not drawing the vertical and horizontal axes at the same scale, just so that we can kind of visualize this properly. But let's say this is 1, 2, 3, 4, 5, and I could keep going on and on and on. So we see here that when n is equal to 1, a sub n is equal to 1. So this is right over there. So when n is equal to 1, a sub n is equal to 1. So this is y is equal to a sub n. Now, when n is equal to 2, we have a sub n is equal to negative 1/2. When n is equal to 3, a sub n is equal to 1/3, which is right about there. When n is equal to 4, a sub n is equal to negative 1/4, which is right about there. And then when n is equal to 5, a sub n is equal to positive 1/5, which is maybe right over there. And we keep going on and on and on. So you see the points, they kind of jump around, but they seem to be getting closer and closer and closer to 0. Which would make us ask a very natural question-- what happens to a sub n as n approaches infinity? Or another way of saying that is, what is the limit-- let me do this in a new color-- of a sub n as n approaches infinity? Well, let's think about if we can define a sub n explicitly. So we can define this sequence as a sub n where n starts at 1 and goes to infinity with a sub n equaling-- what does it equal? Well, if we ignore sign for a second, it looks like it's just 1 over n. But then we seem like we oscillate in signs. We start with a positive, then a negative, positive, negative. So we could multiply this times negative 1 to the-- let's see. If we multiply it times negative 1 to the n, then this one would be negative and this would be positive. But we don't want it that way. We want the first term to be positive. So we say negative 1 to the n plus 1 power. And you can verify this works. When n is equal to 1, you have 1 times negative 1 squared, which is just 1, and it'll work for all the rest. So we could write this as equaling negative 1 to the n plus 1 power over n. And so asking what the limit of a sub n as n approaches infinity is equivalent to asking what is the limit of negative 1 to the n plus 1 power over n as n approaches infinity is going to be equal to? Remember, a sub n, this is just a function of n. It's a function where we're limited right over here to positive integers as our domain. But this is still just a limit as something approaches infinity. And I haven't rigorously defined it yet, but you can conceptualize what's going on here. As n approaches infinity, the numerator is going to oscillate between positive and negative 1, but this denominator is just going to get bigger and bigger and bigger and bigger. So we're going to get really, really, really, really small numbers. And so this thing right over here is going to approach 0. Now, I have not proved this to you yet. I'm just claiming that this is true. But if this is true-- so let me write this down. If true, if the limit of a sub n as n approaches infinity is 0, then we can say that a sub n converges to 0. That's another way of saying this right over here. If it didn't, if the limit as n approaches infinity didn't go to some value right here-- and I haven't rigorously defined how we define a limit-- but if this was not true, if we could not set some limit-- it doesn't have to be equal to 0. As long as it-- if this was not equal to some number, then we would say that a sub n diverges.
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 harmonic series):
- The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"; see divergence of the sum of the reciprocals of the primes):
- 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"):
- The reciprocals of powers of any n>1 produce a convergent series:
- Alternating the signs of reciprocals of powers of 2 also produces a convergent series:
- Alternating the signs of reciprocals of powers of any n>1 produces a convergent series:
- The reciprocals of Fibonacci numbers produce a convergent series (see ψ):
Convergence tests
There are a number of methods of determining whether a series converges or diverges.
Comparison test. The terms of the sequence are compared to those of another sequence . If, for all n, , and converges, then so does
However, if, for all n, , and diverges, then so does
Ratio test. Assume that for all n, is not zero. Suppose that there exists such that
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:
- where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
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 monotonically decreasing function. If
then the series converges. But if the integral diverges, then the series does so as well.
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 monotonically 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.
Conditional and absolute convergence
For any sequence , for all n. Therefore,
This means that if converges, then also converges (but not vice versa).
If the series converges, then the series is absolutely convergent. The Maclaurin series of the exponential function is absolutely convergent for every complex value of the variable.
If the series converges but the series diverges, then the series is conditionally convergent. The Maclaurin series of the logarithm function is conditionally convergent for x = 1.
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.
Uniform convergence
Let be a sequence of functions. The series is said to converge uniformly to f if the sequence of partial sums defined by
converges uniformly to f.
There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test.
Cauchy convergence criterion
The Cauchy convergence criterion states that a series
converges if and only if the sequence of partial sums is a Cauchy sequence. This means that for every there is a positive integer such that for all we have
This is equivalent to
See also
External links
- "Series", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Weisstein, Eric (2005). Riemann Series Theorem. Retrieved May 16, 2005.