5 Sequences

A structured guide to defining sequences, evaluating their limits, and proving convergence or divergence using limit laws, bounds, monotonicity, the Squeeze Theorem, and recursive methods.

Sequences and Their Definitions

A is an ordered list of real numbers indexed by the positive integers:

\{a_n\}_{n=1}^{\infty}=a_1,a_2,a_3,\ldots

The index $n$ is an integer, and $a_n$ denotes the $n$ th term. A can be given explicitly, such as

a_n=\frac{2n+1}{n+3},

or recursively, by specifying initial values and a rule for producing later terms. For example,

a_1=1,\qquad a_{n+1}=\frac{a_n+3}{2}.

An explicit formula calculates a term directly from its index. A recursive rule calculates each new term from previously known terms, so the initial information is essential.

Takeaway: A is a discrete function whose inputs are positive integers, and its terms may be defined directly or through a recurrence.

Limits of Sequences

A to a real number $L$ when its terms become arbitrarily close to $L$ for sufficiently large indices. The notation is

\lim_{n\to\infty}a_n=L \qquad\text{or}\qquad a_n\to L.

Formally, convergence means that for every $\varepsilon>0$ , there is an integer $N$ such that

\lvert a_n-L\rvert<\varepsilon \qquad\text{whenever }n\ge N.

The integer $N$ may depend on $\varepsilon$ . Once $n\ge N$ , every term lies in the interval $(L-\varepsilon,L+\varepsilon)$ .

If no finite real number satisfies this condition, the . For example,

\frac{1}{n}\to0,

whereas $(-1)^n$ because its terms alternate between $1$ and $-1$ . The $n^2$ also because its terms grow without bound. The notation $a_n\to\infty$ describes unbounded growth rather than convergence to a finite real number.

Takeaway: Convergence concerns the eventual behavior of all terms, not whether the is merely or whether some terms are close to a particular value.

Computing Limits

When a has the form $a_n=f(n)$ , a known function limit can often be used. If

\lim_{x\to\infty}f(x)=L,

then

\lim_{n\to\infty}a_n=L.

For example,

\lim_{n\to\infty}\frac{3n^2-1}{n^2+4} =\lim_{n\to\infty}\frac{3-1/n^2}{1+4/n^2}=3.

Frequently useful limits include

\lim_{n\to\infty}\frac{1}{n^p}=0\quad(p>0), \qquad \lim_{n\to\infty}r^n=0\quad(\lvert r\rvert<1), \qquad \lim_{n\to\infty}\left(1-\frac{1}{n}\right)=1.

If $a_n\to L$ and $b_n\to M$ , then the limit laws give

\lim_{n\to\infty}(a_n+b_n)=L+M,

\lim_{n\to\infty}(a_n-b_n)=L-M,

\lim_{n\to\infty}a_nb_n=LM,

and, when $M\ne0$ ,

\lim_{n\to\infty}\frac{a_n}{b_n}=\frac{L}{M}.

Continuity also transfers limits: if $g$ is continuous at $L$ , then

\lim_{n\to\infty}g(a_n)=g(L).

For example, if $a_n\to4$ , then

\lim_{n\to\infty}\sqrt{a_n+5}=\sqrt{9}=3.

Takeaway: Simplify expressions using familiar function limits, algebraic limit laws, and continuity.

The

The is useful when a is difficult to evaluate directly. Suppose that, for all sufficiently large $n$ ,

b_n\le a_n\le c_n,

and both outside sequences have the same limit:

\lim_{n\to\infty}b_n= \lim_{n\to\infty}c_n=L.

Then

\lim_{n\to\infty}a_n=L.

For example, the sine function always satisfies $-1\le\sin n\le1$ . Dividing by the positive number $n$ gives

-\frac{1}{n}\le\frac{\sin n}{n}\le\frac{1}{n}.

Both outside sequences approach zero, so the yields

\lim_{n\to\infty}\frac{\sin n}{n}=0.

The inequalities need only hold from some index onward; finitely many initial terms do not affect a finite limit.

Takeaway: To use the theorem, trap the target between two simpler sequences that approach the same value.

Monotonicity and Boundedness

Monotonicity describes the direction of change between consecutive terms. A is nondecreasing when

a_{n+1}\ge a_n,

and nonincreasing when

a_{n+1}\le a_n.

A is if it is nondecreasing or nonincreasing. To test this property, examine the sign of

a_{n+1}-a_n,

or, for positive terms, compare the ratio $a_{n+1}/a_n$ .

For

a_n=1-\frac{1}{n},

we have

a_{n+1}-a_n =\left(1-\frac{1}{n+1}\right)-\left(1-\frac{1}{n}\right) =\frac{1}{n(n+1)}>0,

so the is increasing.

Monotonicity alone does not guarantee convergence. The $a_n=n$ is increasing but has no finite limit because it is unbounded. A bound in the appropriate direction is also needed.

A is above if some number $U$ satisfies $a_n\le U$ for every $n$ . It is below if some number $L$ satisfies $a_n\ge L$ for every $n$ . It is when both conditions hold. Every convergent is , but a need not converge; $(-1)^n$ is a counterexample.

Takeaway: Monotonicity controls direction, while boundedness prevents continued escape. Together, in the correct directions, they establish convergence.

Proving Convergence for Recursive Sequences

The provides a practical convergence test:

Every increasing that is above .
Every decreasing that is below .

This theorem is especially useful for recursive sequences, whose general terms may be difficult to express explicitly.

Consider

a_1=1, \qquad a_{n+1}=\sqrt{2+a_n}.

First, prove the upper bound $a_n\le2$ . The claim is true for $a_1=1$ . If $a_n\le2$ , then

a_{n+1}=\sqrt{2+a_n}\le\sqrt{4}=2.

Therefore, induction proves that every term is at most $2$ .

Next, prove that the is increasing. Since $a_1\le a_2$ , suppose $a_n\ge a_{n-1}$ . Because the square-root function is increasing,

a_{n+1}=\sqrt{2+a_n} \ge\sqrt{2+a_{n-1}}=a_n.

Thus the is increasing and above, so it . Write its limit as $L$ . Taking limits in the recurrence gives the

L=\sqrt{2+L}.

Since every term is nonnegative, $L\ge0$ . Squaring produces

L^2=2+L, \qquad L^2-L-2=0, \qquad (L-2)(L+1)=0.

The algebraic candidates are $L=2$ and $L=-1$ . The bound $L\ge0$ rejects $L=-1$ , so

\lim_{n\to\infty}a_n=2.

The supplies only possible limits. A separate convergence argument is required before taking limits in the recurrence, and bounds or other properties must be used to discard incompatible algebraic solutions.

Takeaway: For a , a reliable order of work is to compute initial terms, prove a bound, prove monotonicity, apply the , and then solve the resulting .

Convergence and Divergence Strategies

Several criteria organize decisions about convergence and divergence:

If $a_n\to L$ , then the is .
A with a bound in the appropriate direction .
If subsequences approach different values, the full cannot converge. For example, the even and odd subsequences of $(-1)^n$ approach different values.
A whose terms become arbitrarily large in magnitude cannot converge to a finite real number.
Changing finitely many initial terms does not change convergence or the value of a finite limit.

These criteria should be applied strategically. Direct algebra is often best for a rational expression, the is useful for oscillatory expressions with shrinking magnitude, and monotonicity plus boundedness is particularly effective for recursively defined terms.

Final summary: A is an ordered list indexed by positive integers. To analyze it, identify its definition, determine whether its terms approach a single value, and choose an appropriate proof method. Limit laws evaluate many explicit formulas; the handles useful inequalities; and the combination of monotonicity and boundedness proves convergence even when no explicit formula is available.