Sequences, series and limits
Sequences and series questions at this level mix algebra with limiting behaviour. You should recognise arithmetic and geometric patterns, solve first-order recurrences, prove statements by induction, and decide whether an infinite series converges.
Core rules
- Arithmetic: a_n=a_1+(n-1)d, S_n=n(a_1+a_n)/2. Geometric: a_n=a_1 r^{n-1}, S_n=a_1(1-r^n)/(1-r) for r≠1.
- An infinite geometric series converges exactly when |r|<1, with sum a_1/(1-r).
- For a recurrence a_{n+1}=f(a_n), a finite limit L must satisfy L=f(L); then prove convergence or find a closed form.
- Induction: prove the base case, assume the statement for n=k, and derive it for n=k+1 using that assumption.
- Telescoping and partial fractions turn many series into a few surviving terms.
Worked example
Let a_1=0 and a_{n+1}=(a_n+2)/3. Look for a fixed point: L=(L+2)/3 gives 3L=L+2, so L=1. Put b_n=a_n-1. Then b_{n+1}=a_{n+1}-1=(a_n+2)/3-1=(a_n-1)/3=b_n/3. Since b_1=-1, b_n=-(1/3)^{n-1}, so a_n=1-(1/3)^{n-1}. As n grows, (1/3)^{n-1} tends to 0, hence a_n tends to 1. The closed form also proves the limit rather than merely guessing it.
Traps
Do not apply a geometric sum formula when r=1. In induction, the assumption for k must actually be used; proving the next case independently is not induction. For recurrences, a fixed point is only a candidate limit until convergence is justified. Watch index shifts in sums and do not confuse the nth term with the sum of n terms.