Sequences of numbers appear in diverse situations. If you divide a cake in half, and then divide the remaining half in half, and continue dividing in half indefinitely, then the fraction of cake remaining at each step forms the sequence This is the sequence of values of the function

Formally, a sequence is an ordered collection of numbers defined by a function f (n) on a set of integers. The values a n = f (n) are called the terms of the sequence, and n is called the index. Informally, we think of a sequence {a n } as a list of terms: The sequence does not have to start at n = 1, but may start at n = 0, n = 2, or any other integer. When a n is given by a formula, we refer to a n as the general term.

Compute a 2, a 3, a 4 for the sequence defined recursively by The sequence in the next example is defined recursively. The first term is given and the nth term a n is computed in terms of the preceding term a n-1.

Our main goal is to study convergence of sequences. A sequence {a n } converges to a limit L if |a n − L| becomes arbitrary small when n is sufficiently large. Here is the formal definition. DEFINITION Limit of a Sequence We say that {a n } converges to a limit L, and we write if, for every ε > 0, there is a number M such that |a n − L| M. If no limit exists, we say that {a n } diverges. If the terms increase without bound, we say that {a n } diverges to infinity. Proof

If {a n } converges, then its limit L is unique. A good way to visualize the limit is to plot the points (1, a 1 ), (2, a 2 ), (3, a 3 ),…. The sequence converges to L if, for every ε > 0, the plotted points eventually remain within an ε-band around the horizontal line y = L.

The figure below shows the plot of a sequence converging to L = 1.

On the other hand, we can show that the sequence a n = cos n has no limit.

Solution The definition requires us to find, for every ε > 0, a number M such that Formal Definition of Limit

Note the following two facts about sequences: The limit does not change if we change or drop finitely many terms of the sequence. If C is a constant and a n = C for all n sufficiently large, then Many of the sequences we consider are defined by functions; that is, a n = f (n) for some function f (x). For example, A fact we will use often is that if f (x) approaches a limit L as x → ∞, then the sequence a n = f (n) approaches the same limit L. Indeed, for all ε > 0, we can find M so that |f (n) − L| M. It follows automatically that |f (n) − L| M.

If f (x) converges to L, then the sequence a n = f (n) also converges to L.

THEOREM 1 Sequence Defined by a Function Find the limit of the sequence

THEOREM 1 Sequence Defined by a Function

The limit of the Balmer wavelengths b n in the next example plays a role in physics and chemistry because it determines the ionization energy of the hydrogen atom. Table 1 suggests that b n approaches Figure 6 shows the graph, and in Figure 7, the wavelengths are shown “crowding in” toward their limiting value.Table 1Figure 6Figure 7 Calculate the limit of the Balmer wavelengths

A geometric sequence is a sequence a n = cr n, where c and r are nonzero constants. Each term is r times the previous term; that is, a n /a n-1 = r. The number r is called the common ratio. For instance, if r = 3 and c = 2, we obtain the sequence (starting at n = 0) In the next example, we determine when a geometric series converges. Recall that {a n } diverges to ∞ if the terms a n increase beyond all bounds; that is, for every number N, a n > N for all sufficiently large n If r > 1, the geometric sequence a n = r n diverges to ∞.

Geometric Sequences with r ≥ 0 Prove that for r ≥ 0 and c > 0, f (x) = cr x. If 0 ≤ r < 1, then If r > 1, then both f (x) and the sequence {cr n } diverge to ∞ (because c > 0). If r = 1, then cr n = c for all n, and the limit is c.

THEOREM 2 Limit Laws for Sequences Assume that {a n } and {b n } are convergent sequences with Then:

THEOREM 3 Squeeze Theorem for Sequences Let {a n }, {b n }, {c n } be sequences such that for some number M,

Geometric Sequences with r < 0 Prove that for c If -1 < r < 0, then If r = -1, then If r < -1, then

As another application of the Squeeze Theorem, consider the sequence Both the numerator and the denominator grow without bound, so it is not clear in advance whether {a n } converges. The graph & table suggest that a n increases initially and then tends to zero. In the next example, we verify that a n = R n /n! converges to zero for all R. This fact is used in the discussion of Taylor series in Section 10.7.Section 10.7

Let M be a positive integer such that Since CR/n → 0, the Squeeze Theorem gives us The first M factors are ≥ 1 and the last n − M factors are < 1. If we lump together the first M factors and call the product C, and drop all the remaining factors except the last factor R/n, we see that For n > M, we write R n /n! as a product of n factors: as claimed. If R < 0, the limit is also zero by Example 7 because |R n /n!| tends to zero.Example 7

Given a sequence {a n } and a function f (x), we can form the new sequence f (a n ). It is useful to know that if f (x) is continuous and a n → L, then f (a n ) → f (L). THEOREM 4 If f (x) is continuous and In other words, we may “bring a limit inside a continuous function.”

Apply Theorem 4 to the sequence and to the functions (a) f (x) = e x and (b) g (x) = x 2. With f (x) = e x we have According to Theorem 4, With g(x) = x 2 we have, and according to Theorem 4,

Of great importance for understanding convergence are the concepts of a bounded sequence and a monotonic sequence. DEFINITION Bounded Sequences A sequence {a n } is: Bounded from above if there is a number M such that a n ≤ M for all n. The number M is called an upper bound. Bounded from below if there is a number m such that a n ≥ m for all n. The number m is called a lower bound. The sequence {a n } is called bounded if it is bounded from above and below. A sequence that is not bounded is called an unbounded sequence.

Upper and lower bounds are not unique. If M is an upper bound, then any larger number is also an upper bound, and if m is a lower bound, then any smaller number is also a lower bound (Figure 11).Figure 11

As we might expect, a convergent sequence {a n } is necessarily bounded because the terms a n get closer and closer to the limit. This fact is recorded in the next theorem. THEOREM 5 Convergent Sequences Are Bounded If {a n } converges, then {a n } is bounded.

There are two ways that a sequence {a n } can diverge. One way is by being unbounded. For example, the unbounded sequence a n = n diverges: 1, 2, 3, 4, 5, 6,… However, a sequence can diverge even if it is bounded. This is the case with a n = (−1) n+1, whose terms a n bounce back and forth but never settle down to approach a limit: 1, −1, 1, −1, 1, −1,… There is no surefire method for determining whether a sequence {a n } converges, unless the sequence happens to be both bounded and monotonic. By definition, {a n } is monotonic if it is either increasing or decreasing: {a n } is increasing if a n < a n+1 for all n. {a n } is decreasing if a n > a n+1 for all n.

Intuitively, if {a n } is increasing and bounded above by M, then the terms must bunch up near some limiting value L that is not greater than M (Figure 12).Figure 12

THEOREM 6 Bounded Monotonic Sequences Converge If {a n } is increasing and a n ≤ M, then {a n } converges and If {a n } is decreasing and a n ≥ m, then {a n } converges and

Show that the following sequence is bounded and increasing:

Prove that Step 1. Show that {a n } is bounded above. We claim that M = 2 is an upper bound. We certainly have a 1 < 2 because On the other hand, This is true because Now, since a 1 < 2, we can apply to conclude that a 2 < 2. Similarly, a 2 < 2 implies a 3 < 2, and so on for all n. Formally speaking, this is a proof by induction. Step 2. Show that {a n } is increasing. Since a n is positive and a n < 2, we have This shows that {a n } is increasing. We conclude that the limit L exists and hence L = 2.