Sequences, Series and Limits
Core concepts
You must handle arithmetic and geometric sequences, their nth terms and partial sums, infinite geometric series, and the idea of a limit. Know that an arithmetic sequence has constant difference d with a_n = a_1 + (n-1)d and S_n = n(a_1 + a_n)/2, while a geometric sequence has constant ratio r with a_n = a_1 r^(n-1) and S_n = a_1(1 - r^n)/(1 - r). An infinite geometric series converges to a_1/(1 - r) only when |r| < 1. For limits, understand behaviour as n grows, dominant terms, and simple rational-function limits.
Traps to watch
- The infinite geometric sum formula requires |r| < 1; otherwise the series diverges.
- Sum formulas use n terms, not the value of the last term.
- In limits of rational functions, divide by the highest power of n; if degrees match, the limit is the ratio of leading coefficients.
- Alternating signs mean r is negative - check convergence with |r|.
- Confusing the nth term with the sum of the first n terms.
Rule of thumb
For limits at infinity, compare degrees: numerator degree lower gives 0, equal gives leading-coefficient ratio, higher gives divergence.
Worked example
Find the sum of the infinite series 3 + 1 + 1/3 + 1/9 + ... Here a_1 = 3 and r = 1/3, so |r| < 1 and S = 3/(1 - 1/3) = 3/(2/3) = 4.5. If the series were 3 + 6 + 12 + ..., r = 2 and the series diverges, so no finite sum exists.