Approximating Infinite Series: Calculating Sum and Estimating Error

[physstudent1]

Homework Statement

The infinite Series starts at n=1 and is (4-sin(n))/(n^2 + 1)

For each series which converges, give an approximation of its su, together with an error estimate, as follows. First calculate the sum s_5 of the first 5 terms, Then estimate the "tail" which is the infinite series starting at n=6 by comparing it with an appropiate improper integral or geometric series.

Homework Equations

The Attempt at a Solution

Ok, so to start off I proved it converges by comparing it to 5/n^2 since this series is larger then the original one and it converges by the p-series test then the original series also converges. I calculated the first five sums and got 2.863 for my s_5 I'm unsure of how to calculate the tail however, and unsure of how to calculate the error. At first I was thinking to do the improper integral from 6 to infinite of 5/(n^2) since I compared it to this before, but with that I got .8 and that seemed large for the tail of this series. I am also unsure of how to find the error? I was thinking that once i find the value of the tail that the value of (s_5 + tail) - (s_5) would be the error? I don't really know. Please help thanks!

 

[Gib Z]

I vaguely remember posting to the same question a few days ago. Was my maximum error just too high?

 

[physstudent1]

I saw this but I just don't understand how you actually found the error...

 

[physstudent1]

are the tail value and the error the same thing?

 

[Gib Z]

The maximum tale value is the maximum error, yes.