MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series
Problem
For a positive parameter k, consider the following two infinite geometric series.
Series M: first term k and common ratio 21.
Series N: first term 6 and common ratio 12k.
Address the questions below.
(i) Write the condition on k for which each of the two series converges.
(ii) Under the convergence conditions obtained in part (i), set up an inequality that expresses the requirement that the sum to infinity of Series M exceeds that of Series N.
(iii) Solve the inequality derived in part (ii) to obtain the range of values of k for which the stated requirement is satisfied.