Sequences

A sequence is a function whose domain is the set of positive integers:

is called the nth term, and the sequence is denoted by .

______

Ex. Write the first five terms of the sequence.

(a)

(b)

______

Ex. Write an expression for the nth term.

(a) 3, 8, 13, 18, …

(b)

(c) 1, 4, 9, 16, 25, …

(d) 4, 10, 28, 82, …

(e)

______

If a sequence has a finite limit as , we say that the sequence converges to that limit.

If a sequence does not have a finite limit as , then we say that the sequence diverges.

Ex. Determine whether the following sequences converge or diverge. If the sequence converges,

find its limit. If it diverges, show why.

(a)

(b)

(c)

(d)

(e)

Geometric Series Test and nth Term Test for Divergence

Geometric Series Test
A geometric series is in the form
The geometric series diverges if .
If the series converges to the sum

Ex. Determine whether the following series converge or diverge.

(a)

(b)

______

nth Term Test for Divergence
If , then the series diverges.

Note: This does NOT say that if , then the series converges. This test can only

be used to prove divergence. If , then this test doesn’t tell us anything, and

we need to use another test.

Ex. Determine whether the following series converge or diverge.

(a)

(b)

(c)

Integral Test and p-Series

Integral Test If f is positive, continuous, and decreasing for and either both converge or both diverge.

Ex. Determine whether the following series converge or diverge.

(a)

(b)

p-Series
A series of the form is called a p-series, where p is a positive constant. For p = 1, the series is called the harmonic series.
p-Series Test
The p-series
a) converges if p > 1
b) diverges if p < 1
c) diverges if p = 1 (harmonic series)

Ex. Determine whether the following series converges or diverges.

Comparison of Series

Direct Comparison Test
If
1) If ______.
2) If ______.

Ex. Determine whether the following converge or diverge.

(a)

(b)

(c)

______

Limit Comparison Test
Suppose where L is both finite and positive.
Then the two series either both converge or both diverge.

Ex. Determine whether the following converge or diverge.

(a)

(b)

Alternating Series

An alternating series is a series whose terms are alternately positive and negative.

Examples:

In general, just knowing that tells us very little about the convergence of the

series ; however, it turns out that an alternating series must converge if the terms have a limit of 0 and the terms decrease in magnitude.

Alternating Series Test
If , then an alternating series
converges if both of the following conditions are satisfied:
1)
2) is a decreasing sequence; that is, for all n.

Note: This does notsay that if , the series diverges by the Alternating Series

Test. The Alternating Series Test can only be used to prove convergence.

If , then the series diverges by the nth Term Test for Divergence,

not by the Alternating Series Test.

______

Ex. Determine whether the following series converge or diverge.

(a)

(b)

Alternating Series Remainder
Suppose a series has terms that are alternating, decreasing in magnitude, and having a limit of 0. If the series has a sum S, then , where is the nth partial sum of the series.
In other words, if the three conditions are met, you can approximate the sum of the series by using the nth partial sum, , and your error will be bounded by the first truncated term,

Ex. Approximate the sum, S, of the series by using its first four terms, and explain

why your estimate differs from the actual value by less than Then use your results to

find an interval in which S must lie.

Ex. How many terms are needed to approximate the sum of the series so that the

estimate differs from the actual sum by less than Justify your answer.

Ratio Test

Ratio Test
Let be a series of nonzero terms.
1) converges if .
2) diverges if .

Ex. Determine whether the following converge or diverge.

(a)

(b)

(c)