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CAIE A-Level Further Math 1.3.3 Series convergence

Practise deciding when a series converges by using the sum to n terms and evaluating the limiting sum where it exists.

Syllabus
2028–2030
Course
Further Mathematics 9231
Level
AS

Exam points

  • take the limit of a telescoped sum to n terms to state a sum to infinity
  • use a finite-sum result to deduce an infinite-series value without restarting the calculation

1.3.3—Series convergence question 1

[Maximum number: 1]

Deduce the value of ∑r=1∞3r2+3r+1r3(r+1)3\sum_{r=1}^{\infty} \frac{3 r^{2}+3 r+1}{r^{3}(r+1)^{3}}.

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