Sequence limits: lim(n->infinity) a_n where n is a natural number. All limit laws apply. Key results:

• lim(n->infinity) 1/n^k = 0 for k > 0
• lim(n->infinity) r^n = 0 if |r| < 1, = 1 if r = 1, diverges if |r| > 1
• lim(n->infinity) n^(1/n) = 1
• lim(n->infinity) (1/n!) = 0
• lim(n->infinity) (n!)^(1/n) = infinity

Sum-to-integral conversion (Riemann sums): lim(n->infinity) (1/n) * sum(r=0 to n-1) f(r/n) = integral from 0 to 1 of f(x) dx.

Example: lim(n->infinity) [1/(n+1) + 1/(n+2) + ... + 1/(2n)] = lim (1/n) * sum(r=1 to n) 1/(1 + r/n) = integral from 0 to 1 of 1/(1+x) dx = ln(2).

Recursive sequences: If a_(n+1) = f(a_n) and the sequence converges to L, then L = f(L). Solve this equation to find L. But first verify convergence (e.g., show the sequence is monotonic and bounded).

JEE Pattern: Riemann sum problems appear regularly. The key is recognizing the pattern (1/n) * sum f(r/n) and converting to an integral.