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^$\frac{1}{n}$ = 1
• lim(n->infinity) (1/n!) = 0
• lim(n->infinity) (n!)^$\frac{1}{n}$ = infinity

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

Example: lim(n->infinity) [$\frac{1}{n+1}$ + $\frac{1}{n+2}$ + ... + $\frac{1}{2n}$] = lim $\frac{1}{n}$ * sum(r=1 to n) $\frac{1}{1 + r/n}$ = integral from 0 to 1 of $\frac{1}{1+x}$ 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 $\frac{1}{n}$ * sum f$\frac{r}{n}$ and converting to an integral.