Sequences, Series, and Limits
This domain tests ε-N convergence, monotone and Cauchy criteria, subsequences and limsup, and the convergence tests for series with their exact hypotheses. Most lost marks come from applying a test outside its range of validity rather than from arithmetic.
Convergence and structural results
A sequence converges to L when for every ε > 0 there is N with |a_n − L| < ε for all n ≥ N. A bounded monotone sequence converges. Bolzano–Weierstrass: every bounded real sequence has a convergent subsequence. The Cauchy criterion says (a_n) converges iff for every ε > 0 there is N with |a_m − a_n| < ε whenever m, n ≥ N. Also limsup a_n = lim_{n} sup_{k≥n} a_k, with liminf defined analogously; convergence is equivalent to finite limsup = liminf.
Series
A series converges iff its partial sums form a Cauchy sequence. Hence a_n → 0 is necessary but not sufficient. For nonnegative terms use comparison, limit comparison, ratio, root, integral, and Cauchy condensation tests. If the ratio or root limit is L, the series converges for L < 1, diverges for L > 1, and the test is inconclusive at L = 1. The Leibniz test applies when b_n decreases to 0: then the alternating series converges, with error bounded by the first omitted term. Absolute convergence implies convergence; a conditionally convergent series can be rearranged to converge to any prescribed sum.
Worked example
Test the series with a_n = n!/n^n, whose terms are positive. Compute the ratio: a_{n+1}/a_n = [(n+1)!/(n+1)^{n+1}]·[n^n/n!] = (n/(n+1))^n = 1/(1 + 1/n)^n. Since (1 + 1/n)^n → e, this ratio tends to 1/e, which is less than 1. The ratio test therefore gives convergence.
Traps
Concluding convergence because a_n → 0; quoting a ratio-test conclusion when the limit equals 1; reversing the inequality in the ratio test; using comparison on series with negative terms; assuming termwise manipulation is valid for conditionally convergent series; forgetting that the Leibniz test requires monotone decrease to zero.