Sequences and Series
What you must understand
You need arithmetic and geometric sequences, sums to (n) terms, limiting sums of infinite geometric series, and the behaviour of sequences as (n\to\infty). Extension 2 adds telescoping series, partial fractions to sum series, the method of differences, and convergence tests in an informal sense (comparison, ratio, recognising divergence). You should also handle recurrence relations, prove closed forms by induction, and work with series derived from binomial expansions.
Traps
- Limiting sum condition: (S_\infty = \frac{a}{1-r}) requires (|r| < 1); applying it when (|r|\ge 1) is invalid.
- Off-by-one in (n): check whether the series has (n) or (n+1) terms, especially when starting at (k=0).
- Telescoping signs: write out several terms to confirm which cancel; a sign slip leaves a phantom term.
- Partial fractions before summing: split (\frac{1}{k(k+1)}) into (\frac{1}{k} - \frac{1}{k+1}) before telescoping.
- Divergence test misuse: a term tending to zero does not guarantee convergence, though a term not tending to zero guarantees divergence.
Rule of thumb
For a series with rational terms, attempt partial fractions first; if consecutive terms then share structure, telescope and evaluate the surviving first and last terms.
Worked example
Sum (\sum_{k=1}^{n} \frac{1}{k(k+1)}). Since (\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}), the sum telescopes: [ \left(1 - \tfrac12\right) + \left(\tfrac12 - \tfrac13\right) + \cdots + \left(\tfrac1n - \tfrac{1}{n+1}\right) = 1 - \frac{1}{n+1}. ] As (n\to\infty) the sum tends to 1.