Advanced Level Further Pure Mathematics: Sequences and Series Notes

Advanced Level Further Pure Mathematics: Sequences and Series

A sequence is an ordered list of numbers, usually written (u_n). It may be defined explicitly by a formula for u_n, or recursively by giving a starting value and a rule linking consecutive terms.

1. Explicit and recursive definitions

An explicit sequence has the form u_n = f(n). For example, u_n = 3n − 2 gives 1, 4, 7, 10, … when n begins at 1. A recursive sequence such as u₀ = 2 and u_n+1 = ½u_n + 3 is generated one term at a time.

2. Monotonicity

A sequence is increasing if u_n+1 ≥ u_n, decreasing if u_n+1 ≤ u_n, and strictly monotone when the inequality is strict. To study monotonicity, examine u_n+1 − u_n or, for positive terms, u_n+1/u_n.

Example: For u_n = n/(n+1), u_n+1 − u_n = 1/[(n+1)(n+2)] > 0. Therefore the sequence is strictly increasing.

3. Bounded sequences

A sequence is bounded above if u_n ≤ M for every n, and bounded below if u_n ≥ m. If both conditions hold, the sequence is bounded. A monotone increasing sequence that is bounded above converges; a monotone decreasing sequence that is bounded below also converges.

4. Limits and convergence

If u_n approaches a finite number L as n becomes arbitrarily large, write lim u_n = L. Standard strategies include dividing numerator and denominator by the highest power of n, using known limits, applying the squeeze theorem and comparing dominant terms.

5. Graphical representation of a recurrence

For u_n+1 = f(u_n), draw y=f(x) together with y=x. Starting at u₀ on the x-axis, move vertically to y=f(x), horizontally to y=x, and repeat. The resulting staircase or cobweb diagram shows whether terms approach or move away from a fixed point.

6. Arithmetic sequences and series

For first term a and common difference d:

u_n = a + (n−1)d
S_n = n/2[2a + (n−1)d] = n/2(a + l)

where l is the final term.

7. Geometric sequences and series

For first term a and common ratio r:

u_n = arⁿ⁻¹
S_n = a(1−rⁿ)/(1−r), r ≠ 1.

If |r| < 1, the infinite series converges to S_∞ = a/(1−r).

8. Tests and telescoping

If the terms of an infinite series do not tend to zero, the series diverges. For a telescoping series, use partial fractions or rearrangement so that most terms cancel. Always write several terms before stating the remaining first and last terms.

Worked example

Let u₀=2 and u_n+1=(u_n+6)/3. A possible limit satisfies L=(L+6)/3, so L=3. Also u_n+1−3=(u_n−3)/3, hence u_n−3=−1/3ⁿ. Therefore u_n=3−3⁻ⁿ and u_n→3.

Practice questions

1. Show that u_n=(2n+1)/(n+2) is increasing and bounded above.
2. For u₀=4 and u_n+1=√(2u_n+3), conjecture and prove the limit.
3. Find the sum of the first 20 terms of 7, 11, 15, …
4. Determine the sum to infinity of 12−6+3−3/2+…
5. Evaluate Σ_r=1ⁿ1/[r(r+1)] and hence find its limit.