CAIE A-Level Further Math AS 1.3.3 Series Convergence Questions

Determine whether a series converges by taking the limit of its partial sums, then find the sum to infinity and check any parameter boundaries separately.

Exam points

use the finite partial-sum expression to establish a finite limit as the number of terms tends to infinity and deduce the sum to infinity
determine the permitted parameter values from the partial-sum limit and check boundary cases separately, giving the appropriate sum wherever convergence holds

[Maximum number: 1]

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

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CAIE A-Level Further Math AS 1.3.3 Series Convergence Questions question 1