Summation of series — A/L Combined Mathematics

How do I practise Summation of series?

1 question with a worked solution

Check before you answer: Partial Fractions Decomposition: Resolving rational r-th terms accurately into differences of partial fractions (e.g., 1 / [r(r + 1)] = 1/r - 1/(r + 1)).

1. Using the method of differences (telescoping series), what is the sum to n terms S_n = ∑_r=1ⁿ [1 / (r(r + 1))]?
   1. A) 1 / (n + 1)
   2. B) n / (n + 1)
   3. C) (n + 1) / n
   4. D) n / (2n + 1)
   Check my answer

   Correct answer: B) n / (n + 1)

   Expressing in partial fractions: 1 / [r(r + 1)] = 1/r - 1/(r + 1). Summing from r = 1 to n: S_n = (1 - 1/2) + (1/2 - 1/3) + ... + (1/n - 1/(n + 1)). Intermediate terms cancel telescopically, leaving S_n = 1 - 1/(n + 1) = n / (n + 1).

   Show worked solution

   Step 1: Partial fractions: 1 / (r(r+1)) = 1/r - 1/(r+1). Step 2: Telescoping sum: S_n = 1 - 1/(n+1) = n / (n+1). Answer: n / (n + 1).
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Summation of series fit in the syllabus?

Year 1 · Term 3