Sequences, Series, and Limits
Core ideas
Recognize arithmetic and geometric sequences, write closed forms for the nth term and partial sums, and handle telescoping sums and simple recurrences. Understand convergence of infinite geometric series (|r| < 1) and the meaning of a limit informally: what value a sequence or function approaches. Know standard limits such as (1 + 1/n)^n approaching e, and how growth rates compare (log < polynomial < exponential).
Traps to watch
- Using the arithmetic sum formula with the wrong number of terms (count carefully: from m to n inclusive is n - m + 1 terms).
- Applying the infinite geometric sum when |r| >= 1 (it diverges).
- Confusing the nth term with the sum of the first n terms.
- Off-by-one errors in recurrence indexing.
- Assuming a limit exists because terms look small; verify the pattern.
Worked example
Sum 1/(12) + 1/(23) + ... + 1/(n(n+1)). Since 1/(k(k+1)) = 1/k - 1/(k+1), the sum telescopes to 1 - 1/(n+1) = n/(n+1). As n grows, the limit is 1. Telescoping turns a long sum into two surviving boundary terms.
Rule of thumb
For sums of rational terms, try partial fractions to telescope; for geometric sums, identify a, r, and n before plugging into formulas.