Topic 3 of 8 · Form 5
Series and proof by induction
Σr = n(n + 1)/2; Σr² = n(n + 1)(2n + 1)/6; Σr³ = n²(n + 1)²/4.
Notes for this topic 1 file, in order
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Standard results
Σr = n(n + 1)/2; Σr² = n(n + 1)(2n + 1)/6; Σr³ = n²(n + 1)²/4.
Worked example: Show that Σ r(r + 1) from r = 1 to n = n(n + 1)(n + 2)/3.
Σr² + Σr = n(n + 1)(2n + 1)/6 + n(n + 1)/2 = n(n + 1)[(2n + 1) + 3]/6 = n(n + 1)(n + 2)/3
Method of differences
If uᵣ = f(r) − f(r + 1), terms cancel (telescope).
Worked example: Find Σ 1/(r(r + 1)) from r = 1 to n, and the sum to infinity.
1/(r(r + 1)) = 1/r − 1/(r + 1) → sum = 1 − 1/(n + 1) = n/(n + 1); sum to infinity = 1
Maclaurin series (introduction)
f(x) = f(0) + xf′(0) + x²f″(0)/2! + …; e.g. eˣ = 1 + x + x²/2! + …, sin x = x − x³/3! + …, ln(1 + x) = x − x²/2 + x³/3 − … (|x| < 1).
Proof by induction
- Basis: show the statement is true for n = 1.
- Assume true for n = k.
- Show it is then true for n = k + 1.
- Conclude: true for all positive integers n.
Used for series, divisibility (e.g. 7ⁿ − 1 divisible by 6) and matrix powers.
Check yourself
- Evaluate Σr² from r = 1 to 10.
- Write the first three non-zero terms of the Maclaurin series of cos x.
- In an induction proof, what must you show after assuming the statement for n = k?
Show answers
- 10 × 11 × 21/6 = 385
- 1 − x²/2 + x⁴/24
- That the statement is then true for n = k + 1.
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