Important examples

In this section we will cover basic examples of sequences and check on their boundedness and monotonicity. We start with alternating sequence and return to it again at the end, we briefly cover arithmetic sequences, but the most important type is the geometric sequence. We return to these examples again in Theory - Limits - Important examples.

Alternating sequence

An alternating sequence is actually a category, a more general type of a sequence (see the end of this section), but here we will look at the prototype alternating sequence:

The graph looks like this:

From the picture we immediately see that this sequence is bounded (for all n we clearly have |(−1)ⁿ| ≤ 1) and not monotone. Indeed, it is not increasing or non-decreasing because the second term (when n = 1) is less then the first term (when n = 0), so the sequence drops there; and it is not decreasing or non-increasing because the third term is greater than the second term, so the sequence increases there.

It is also possible to index this sequence starting from n = 1. There are two reasons for indexing from zero as we did. The first reason is my whim - I prefer to start with a positive number rather than with −1, and it is actually more convenient this way sometimes. The second reason will become clear when we look at geometric series.

This sequence appears quite often; sometimes by itself, more frequently as a part of another sequence; see the last section below.

Definition.
By an arithmetic sequence we mean any sequence of the form

a_n = a + nd, n = 0,1,2,3,...

where a and d are fixed real numbers.

Thus the arithmetic sequence looks like this:

{a, a + d, a + 2d, a + 3d, a + 4d,...}.

Again, it is possible to index the sequence n = 1,2,3,... but then the first term of the sequence would be a+d, which is not as nice as starting with a.

Note the following: Every term of an arithmetic sequence can be obtained from the previous term by adding the constant d. For instance,

a₅ = a + 5d = (a + 4d) + d = a₄ + d.

This is the main idea of the arithmetic sequence. You start with some number a and then keep adding over and over the same difference d. Indeed, an arithmetic sequence can be alternatively defined in a recursive way like this:
(1)    a₀ = a,
(2)    a_n+1 = a_n + d, n = 0,1,2,3,...

It is not difficult to prove by induction that the explicit and recursive definitions define the same infinite sequence.

One last way to characterize an arithmetic sequence: A given sequence is arithmetic if and only if the difference between successive terms is always the same.

Example:
{2, 0.5, −1, −2.5, −4, −5.5,...} is an arithmetic sequence. Indeed, the difference between successive terms is always d = −1.5. Check that for n = 0,1,2,3,.. the terms are given by a_n = 2 + (−1.5)n.

We chose this example to show that the step d can be also negative. There is a special case when d = 0. Then a_n = a for all n and we get the constant sequence {a, a, a,...}.

What are the properties of arithmetic sequences? First we look at the trivial case of a constant sequence a_n = a for all n.

We immediately see that such a sequence is bounded; moreover, it is monotone, namely it is both non-decreasing and non-increasing.

What happens if d is not equal to zero? There are two cases.

Case 1. If d > 0, then the sequence is increasing (also non-decreasing) as

a_n+1 = a_n + d > a_n.

Also, from a_n = a + nd > a we see that such a sequence is bounded from below. However, it is not bounded from above and hence not bounded.

Case 2. If d < 0, then the sequence is decreasing (also non-increasing) as

a_n+1 = a_n + d < a_n.

Also, from a_n = a + nd < a we see that such a sequence is bounded from above. However, it is not bounded from below (the proof is analogous to the previous case) and hence not bounded.

Definition.
By a geometric sequence we mean any sequence of the form

a_n = aqⁿ, n = 0,1,2,3,...

where a and q are fixed real numbers.

Thus the geometric sequence looks like this (note that aq⁰ = a⋅1 = a):

{a, aq, aq², aq³, aq⁴,...}.

Again, it is possible to index the sequence n = 1,2,3,... but then the first term of the sequence would be aq, which is not as nice as starting with a. In fact, in most cases we have a = 1, so the typical geometric sequence looks like this:

{1, q, q², q³, q⁴,...}.

We already saw one example of a geometric sequence, namely the prototype alternating sequence in the beginning.

Note the following: Every term of a geometric sequence can be obtained from the previous term by multiplying it by the constant q. For instance,

a₅ = aq⁵ = (aq⁴)q = a₄q.

This is the main idea of the geometric sequence. You start with some number and then start multiplying it over and over by the same constant q. Indeed, a geometric sequence can be alternatively defined in a recursive way like this:
(1)    a₀ = a,
(2)    a_n+1 = a_nq, n = 0,1,2,3,...

It is not difficult to prove by induction that the explicit and recursive definitions define the same infinite sequence.

One last way to characterize a geometric sequence: A given sequence is geometric if and only if the ratio of successive terms is always the same.

The example that we saw before (the alternating sequence) is somewhat special, so it is not a good representative of how a geometric sequence behaves. Another exceptional case is when a = q = 1. We obtain the sequence {1, 1, 1, 1, 1,...}. This is a constant sequence that can be also considered an arithmetic sequence. It is actually easy to show by algebra that if a geometric sequence is constant, then necessarily q = 1 and the sequence is also an arithmetic sequence. On the other hand, the only sequences that are both arithmetic and geometric are constant sequences. The appropriate conclusions apply, they are bounded and monotone.
To sum it up, the cases q = −1 and q = 1 are special and every time we deal with a geometric sequence (or a geometric series for that matter), these two will work differently from the rest.

To see what happens when q is a different number we will explore four typical cases, for simplicity we put (as is customary) a = 1.

Example:
The choice q = 2 yields the sequence a_n = 2ⁿ, that is,

{2⁰, 2¹, 2², 2³, 2⁴,...} = {1, 2, 4, 8, 16, 32, 64,...}.

This sequence is increasing, bounded from below (clearly a_n > 0 for all n) but not bounded from above, hence not bounded.

Example:
The choice q = 1/2 yields the sequence a_n = (1/2)ⁿ = 1/2ⁿ, that is,

{1/2⁰, 1/2¹, 1/2², 1/2³, 1/2⁴,...} = {1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64,...}.

This sequence is decreasing, bounded from below (clearly a_n > 0 for all n) but also bounded from above (a_n ≤ 1 for all n = 0,1,2,3,...), hence bounded.

These two examples show the basic two types of a geometric sequence, namely when q > 1 and when 0 < q < 1. In the next two examples we use the same numbers but with the negative sign, thus obtaining the two typical cases for negative q: namely when q < −1 and −1 < q < 0. Since for negative q we can write q = (−1)|q|, you will observe that these next two cases are actually based on the previous two, they are just modified by "attaching signs", the alternating part from our very first example.

Example:
The choice q = −2 yields the sequence a_n = (−2)ⁿ = (−1)ⁿ2ⁿ, that is,

{2⁰, −2¹, 2², −2³, 2⁴,...} = {1, −2, 4, −8, 16, −32, 64,...}.

This sequence is not monotone, not bounded from below and not bounded from above, hence not bounded.

Example:
The choice q = −1/2 yields the sequence a_n = (−1/2)ⁿ = (−1)ⁿ/2ⁿ, that is,

{1/2⁰, −1/2¹, 1/2², −1/2³, 1/2⁴,...} = {1, −1/2, 1/4, −1/8, 1/16, −1/32, 1/64,...}.

This sequence is not monotone, it is bounded from below (clearly a_n > −1 for all n) but also bounded from above (a_n ≤ 1 for all n = 0,1,2,3,...), hence bounded.

For further information see Important examples in Sequences - Theory - Limit.

Note that in the last two graphs we actually just took the graphs of the previous two examples and flipped every second term down around the n-axis. This shows that in investigating sequences with regularly changing signs it may be a good idea to ignore the signs first. This brings us to the last topic.

Our first example was a prototype of an alternating sequence. Now we are ready to look at a more general notion.

Definition.
By an alternating sequence we mean any sequence {a_n} that is of the form a_n = (−1)ⁿb_n for some non-negative real numbers b_n.

By the way, many people would again prefer to start indexing at some even number so that the alternating sequence starts with plus; I think that, say, {b₀, -b₁, b₂, -b₃,...} looks much better. Actually, even easier is to declare that by alternating sequences we also mean sequences whose terms can be written as a_n = (−1)ⁿ⁺¹b_n for some non-negative real numbers b_n. For simplicity we will formally work with the type from the definition, but everything of course works also for this other type with exactly opposite signs.

When investigating an alternating sequence

{a₁, a₂, a₃,...} = {-b₁, b₂, -b₃,...},

it is often easier to just ignore the signs and investigate the sequence {b₁, b₂, b₃,...}, then take the signs into consideration. Indeed, many methods apply only to sequences with positive terms, so they can be used for the sequence without signs but not for the original alternating sequence. Ignoring the sign can also help when using the connection between functions and sequences (see Sequences and functions in Theory - Limits).

In particular, when drawing the graph of an alternating sequence, it is usually easier to first determine the shape of the graph of {b_n}, then add its mirror image around the n-axis and finally draw the terms of the alternating sequence by picking points alternatively from the top and bottom shape:

Note that this idea could be also used to draw graphs of sequences whose sign changes in a regular pattern but not the alternating one. For instance, {1, 2, −4, 8, 16, −32, 64,...} is a sequence (based on our geometric example) that is not alternating, yet the signs have a regular pattern; in such examples we can often disregard minuses where necessary and then take them back into consideration later.

Finally, note that there are sequences that not only are not alternating, but the sign changes do not even follow any pattern; that is, there is no pattern of signs repeated over and over, and even more, by knowing a certain number of terms, it is impossible to determine the next sign just by knowing the signs of those first terms (that is, without explicitly calculating the sign from the formula). Perhaps the easiest example is {sin(n)}, n = 0,1,2,3,... (as usual, the sine function works in radians, not degrees). The sequence goes {sin(1), sin(2), sin(3), sin(4),...}. The first three numbers are positive, but since 4 is greater than π and less than 2π, sin(4) is negative, as is also sin(5) and sin(6), sin(7) is positive and so on. Proving that there is no pattern is rather difficult, you have to take my word for it, but you can get some insight by calculating the first, say, 30 terms. You will see that signs come always in groups of three or four, but the appearance of groups of four is not regular in any way.

Operations with sequences
Back to Theory - Introduction