Sequences and Series

Adding Infinitely Many Terms

A sequence is an ordered infinite list of numbers $a_1, a_2, a_3, \dots$; a series is what you get by adding a sequence's terms: $\sum_{n=1}^{\infty} a_n$. The central question is convergence — does adding infinitely many terms approach a finite total, or grow without bound?

A sequence converges to a limit $L$ if its terms get arbitrarily close to $L$ (e.g. $\frac{1}{n} \to 0$). A series converges if its partial sums $S_N = a_1 + \dots + a_N$ approach a finite limit as $N \to \infty$. Remarkably, an infinite sum of positive terms can be finite: $$\sum_{n=0}^{\infty} \frac{1}{2^n} = 1 + \tfrac12 + \tfrac14 + \tfrac18 + \dots = 2.$$ This is a geometric series $\sum r^n$, which converges precisely when $|r| < 1$, to the neat sum $\frac{1}{1 - r}$ (starting from $n = 0$).

A necessary warning sign: if the terms $a_n$ do not shrink to zero, the series cannot converge (the term test). But shrinking to zero is not sufficient — the classic harmonic series $\sum \frac{1}{n}$ has terms going to zero yet diverges (grows without bound). Convergence is subtler than "terms get small," and mathematicians use tests (comparison, ratio, integral) to decide it.

Series are foundational in computing: they define functions (next lesson's Taylor series), analyze algorithm running times (summing costs across recursion levels), and underlie numerical methods. Knowing whether and how fast a series converges is a practical, not just theoretical, concern.

Common pitfall: assuming that if the terms $a_n \to 0$ the series must converge. That is false — the harmonic series $\sum \frac{1}{n}$ has terms going to zero yet diverges. Terms shrinking to zero is necessary but not sufficient for convergence. (The valid one-way rule is the contrapositive: if the terms do not go to zero, the series definitely diverges.)