1. Numerical series

1. A series is an infinite sum, written . It doesn’t have to start at k = 0.

2. The nth partial sum of the series is

3. The series converges to a number L means: . In other words, for n large enough, is the same as L to any number of decimal places you like.

4. If converges to L, we sometimes write = L . In other words, the sum is as close to L as you want if you take enough terms.

5. Sometimes there is no number L with . In such a case we say that the series diverges. This could happen because the partial sums bounce around, or, in the case of a series like , because the numbers get bigger and bigger without bound.

6. Types of series:

a) is a positive term series means that for all k.

b) An alternating series has terms that switch signs. Every alternating series can be written as either or as where for all k.

c) A series is absolutely convergent if the positive term series is convergent.

If a series is absolutely convergent, it is convergent. The reverse might not be true. For example, if , then converges but is not absolute convergent, since = diverges. There is a name for this situation, as follows:

d) A series is conditionally convergent if converges but diverges.

2. The convergence of the sum of the series

There are two comparison tests. One is called The Comparison Test and the other one is called The Limit Comparison Test. Both tests require one to choose another series to which the given series is compared. During the course of the class, we will learn about certain well known series that you can use in this regard. So far we have learned about the p-series, which includes the Harmonic series. So these comparison tests work well when the given series looks a little like a p-series.

The Comparison Test: Let and be two series of positive terms. Then

1. If converges and for all n sufficiently large, then also converges.

2. If diverges and for all n sufficiently large, then also diverges.

The Limit Comparison Test: Let and be two series of positive terms. If , where C is finite and positive, then either both series converge or both series diverge.

There are several observations that one must keep in mind in applying these Comparison tests.

1. Both tests apply to series of positive terms. Actually, if either of the series have a finite number of negative terms, then the tests still apply because a finite number of terms does not change the convergence or divergence of any series.

2. In general, one will be given a series, say . One then chooses the series from your knowledge of previously studied series. The p-series is often a good choice, but you will need to pick a particular value for p.

3. In applying these tests, one must verify that the conditions of the tests are satisfied. In particular, for the Comparison Test one must verify the appropriate inequality holds.

4. In applying the Limit Comparison Test, if , then no information about convergence or divergence is obtained and you have to choose some other series with which to compare.

5. Neither of these tests provide an estimate of the error as the Integral Test does.

Example 1: Determine whether or not the series converges or diverges.

Solution: (First one must identify a series with which to compare. To this end, notice that the degree of the numerator is 1 and the degree of the denominator is 3 and 3-1=2. This indicates that a good choice of a series for comparison is .) Consider . (The next step is to decide which comparison test to use. For purposes of illustration, both are shown here, but in your homework you need only use one of the tests.)

(Applying the Comparison Test:) Note that converges since it is a constant multiple of the p-series with p=2. Furthermore for all . So by the Comparison Test, the series converges.

(Applying the Limit Comparison Test:) Note that converges since it is a constant multiple of the p-series with p=2. Furthermore . So by the Limit Comparison Test, the series converges.

Example 2: Determine whether or not the series converges or diverges.

Solution: (First one must identify a series with which to compare. To this end, notice that the degree of the numerator is 1 and the degree of the denominator is 3 and 3-1=2. This indicates that a good choice of a series for comparison is .) Consider . (This time the inequality use above does not work because of the minus sign in the denominator. So we use the Limit Comparison Test.) Note that converges since it is a constant multiple of the p-series with p=2. Furthermore . So by the Limit Comparison Test, the series converges.

Example 3: Determine whether or not the series converges or diverges.

Solution: (First one must identify a series with which to compare.) Consider which diverges since it is the Harmonic series. Furthermore for . So by the Comparison Test, the series diverges.

(If one tries to use the Limit Comparison Test, then . So the Limit Comparison Test does not provide an answer for this problem. The Integral Test can be used to show that this series diverges.)

geometric series

A geometric series is the sum of the terms in a geometric sequence. This results from each term being multiplied by a constant value.

e.g. 1 + 2 + 4 + 8 + 16 + … =

e.g. 6 + 2 + =

The sum for a finite geometric series (that is, a geometric series which has a finite number of terms) is s_n = where a₁ is the first term of the sequence, r is the common ratio (or multiplier), and n is the number of terms in the finite series. If the series is infinite, and if |r| < 1, then the infinite series converges to s = .