Question

Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10 Superscript negative 5 in magnitude. Summation from k equals 1 to infinity StartFraction (negative 1 )Superscript k plus 1 Over k cubed EndFraction

Answer 1

Answer:

You need to sum at least 5000 terms

Step-by-step explanation:

Remember what the estimation theorem for alternating series says.

Given a series

$s = \sum\limits_{n=0}^{\infty} (-1)^n a_n$

An upper bound for the error of the series $R_n$  is $a_{n+1}$ so

$R_n = |s-s_n| \leq a_{n+1}$

For this case you want

$\frac{1}{2n+1} \leq \frac{1}{10^4}\\10^4 \leq 2n+1\\(10^4-1 )/2 \leq n$

$(10^4-1 )/2$ is approximately 5000. So you must sum at least 5000 terms.