Convergence and divergence of series — A/L Combined Mathematics

What must I remember about Convergence and divergence of series?

• Sequence of Partial Sums: For an infinite series ∑_r=1^∞ ur, the n-th partial sum is Sn = ∑_r=1ⁿ ur.
• Convergence of Infinite Series: The infinite series converges to sum S (written ∑_r=1^∞ ur = S) if and only if the sequence of partial sums {Sn} converges to a finite limit: lim_n->∞ Sn = S.
• Divergence of Series: If lim_n->∞ Sn does not exist (diverges to +∞, -∞, or oscillates), the series is Divergent and has no finite sum.
• Infinite Geometric Series: S_∞ = a / (1 - r) converges if and only if |r| < 1 (-1 < r < 1); if |r| ≥ 1, the geometric series diverges.
• Necessary condition for convergence (n-th term test): If ∑ ur converges, then lim_n->∞ un = 0; equivalently, if lim_n->∞ un ≠ 0, the series MUST DIVERGE.

Test yourself (1 minute)

1. lim un = 0 does NOT guarantee convergence

   The converse of the n-th term test is FALSE; e.g., harmonic series ∑ (1/n) has lim(1/n) = 0, but diverges to infinity.

2. Sum to infinity of difference series

   If Sn = f(1) - f(n + 1) and lim_n->∞ f(n + 1) = L, then S_∞ = f(1) - L.

3. p-series test

   The infinite p-series ∑_n=1^∞ 1/n^p converges if and only if p > 1; diverges if p ≤ 1.

Practise Convergence and divergence of series questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 20

What should I study first?

Next in the syllabus

Where does Convergence and divergence of series fit in the syllabus?

Year 1 · Term 3