Sequences and Their Limits

A sequence is an ordered list with a rule

A sequence assigns one output to each positive integer. We write it as $a_1,a_2,a_3,\ldots$ or more compactly as $(a_n)$, where $n$ records position in the list. The index is not a continuous input; it advances by whole-number steps. This distinction explains why a sequence may have a limit even though its graph consists of isolated points rather than an unbroken curve.

For the sequence $a_n=1/n$, the first terms are $1,\frac12,\frac13,\frac14,\ldots$. They are positive and get closer to zero as $n$ increases. We express this behavior by writing

$$\lim_{n\to\infty}a_n=0.$$

The notation does not say that an infinite index is substituted into the formula. It describes what happens to the terms when indices become arbitrarily large. No individual positive term equals zero, yet the terms can be made as close to zero as desired by choosing a sufficiently large index. This distinction between “eventually close” and “eventually equal” is foundational for every later convergence test.

Convergence and divergence

A sequence converges when its terms approach one finite number. That number is the sequence’s limit. A sequence diverges when it does not approach a single finite number. Divergence has several forms: $a_n=n$ grows without bound, $a_n=(-1)^n$ alternates between two values, and $a_n=\sin n$ wanders without settling. The word diverge describes the absence of one finite limiting target; it does not mean every divergent sequence becomes infinitely large.

The sequence $a_n=(-1)^n/n$ converges to zero even though its signs alternate. Its magnitude is $1/n$, which shrinks to zero, so the points bounce above and below the target by ever smaller amounts. A table can suggest this behavior, but a table alone cannot prove it. Samples always leave open the possibility that a later pattern changes. The formula, inequalities, and known limit laws supply the justification that numerical evidence cannot.

When $a_n=f(n)$ for a continuous function $f$, a function limit may help: if $\lim_{x\to\infty}f(x)=L$, then $\lim_{n\to\infty}f(n)=L$. For example, $1/n$ inherits the function limit of $1/x$. The reverse direction is not automatically a way to define a function from a sequence; infinitely many different curves can pass through the same discrete points. The sequence has its own behavior and needs its own reasoning.

Limit laws turn structure into conclusions

If $a_n\to A$ and $b_n\to B$, then sums, products, constant multiples, and quotients with nonzero limiting denominator behave as expected. For example,

$$\lim_{n\to\infty}\frac{3n^2-1}{n^2+5} =\lim_{n\to\infty}\frac{3-1/n^2}{1+5/n^2}=3.$$

Dividing the numerator and denominator by the highest power of $n$ exposes the terms that persist. The calculation is not about making $n$ disappear; it compares growth rates. Constants dominate terms like $1/n^2$ because those reciprocal powers become arbitrarily small. The answer $3$ is also plausible from the ratio of leading coefficients.

Some forms need additional thought. The expression $n/(n+1)$ is less than $1$ for every finite $n$, but its limit is $1$. The fact that every term stays below a number does not prevent the sequence from approaching that number. Conversely, a sequence can be bounded yet fail to converge, as $(-1)^n$ shows. Boundedness is useful information, but it is not a convergence proof by itself.

The formal idea of “as close as desired”

The definition $a_n\to L$ says that for every positive tolerance $\varepsilon$, there is an index $N$ such that every term with $n\ge N$ lies within $\varepsilon$ of $L$. Symbolically,

$$|a_n-L|<\varepsilon\quad\text{whenever}\quad n\ge N.$$

The tolerance $\varepsilon$ measures allowed output error, while $N$ tells how far into the list one must go. For $a_n=1/n$ and a tolerance of $0.01$, choosing $N=101$ works because $1/n<0.01$ whenever $n\ge101$. The particular $N$ need not be smallest; it only must guarantee the claimed closeness for all later terms.

This definition is demanding in a productive way. It rules out a sequence that visits a target repeatedly but then keeps leaving it. It also separates an observation from a proof: “the first hundred terms look close” has no statement about every later term, whereas an $N$-based argument does. The language will return in series, where the relevant sequence is not the list of terms but the list of partial sums.

Retrieval check

For $a_n=(2n-3)/(n+4)$, compute the first three terms, predict the long-run value, and justify the limit by dividing by $n$. Then decide whether the sequence is eventually above or below that limit. Finally, explain in one sentence why $a_n\to2$ does not imply that any term must equal $2$. The next lesson turns a sequence of terms into a series of accumulated partial sums, where convergence asks a different question.