Show that the series 3 minus 3 over 4 plus 3 over 9 minus 3 over 25 plus etc is convergent by the alternating series test, and find the number of terms necessary to estimate the sum of...

Question

Show that the series  is convergent by the alternating series test, and find the number of terms necessary to estimate the sum of the series with an error of less than 0.05. 

Answer 1

a_n = (- 1)^{n + 1}3/n²;

error e < Ia_nI or 0.05 < 3/n²; n² > 3/0.05 or n² > 60; n ≥ 8. So we have to take n = 8 terms

This series converges absolutly as p-series with p = 2> 1