Question

Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10 in magnitude. 3k +1 The number of terms that must be summed is (Round up to the nearest integer as needed.)

Answer 1

$\bg_white |a_{k}|=\frac{1}{3k+1}\\ \\ \\|a_{k+1}|\leq 10^{-4}\\ \\=>\frac{1}{3(k+1)+1}\leq 10^{-4}\\ \\=>\frac{1}{3k+4}\leq 10^{-4}\\ \\=>3k+4\geq 10^{4}\\ \\=>3k+4\geq 10000\\ \\=>3k\geq 9996\\ \\=>k\geq \frac{9996}{3}\\ \\=>k\geq 3332\\$

The number of terms that must be summed is 3332