For what value of n will sn be correct to 1 decimal place? Use s10=10/11 to estimate and compare the sum `sum_(n=1)^oo n/(n^2+n^3)`

Expert Answers

An illustration of the letter 'A' in a speech bubbles

We are asked to determine the value for n such that the indicated sum is correct to 1 decimal place.

`S_n=sum_(n=1)^(oo) n/(n^2+n^3)`

The series converges (using the comparison test) and the sum is 1. The formula for the partial sum is 

`sum_(n=1)^m n/(n^2+n^3)=m/(m+1)`

To be accurate to one decimal place we need the partial sum to be greater than or equal to .95:

`m/(m+1)=.95 ==> .05m=.95 ==> m=19`

Thus we need 19 terms for the partial sum to be accurate to one decimal place.

The required value is n=19.

(For n=19 we have the sum of the series equal to .95 which is 1.0 to one decimal place. for n=18 the sum is approximately 0.9473684211 which is 0.9 to one decimal place.)


See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial Team