It would extremely helpful to know the context in which this question occurs. Are you expected to simply find the sum? If that is the case HERE IT IS.How do I find the sum of 1/(1+k^2) from k=1 to 1000? The book didn`t provide any information of how to solve rational sums, nor formulas for greater powers than 1. I have no idea how to do this.
It says: Show that the sum from k=1 to 1000 of 1/(1+k^2) is smaller or equal to 1000.My first thought when I saw the problem was that if it had been something like SUM[1/(k^2 - 1)], it would telescope. But since 1 + k^2 doesn't factor, that won't work. So I was sort of hoping it would be a copying error!
I can't think of any other methods at your level. In general, series are difficult problems.
Again, please quote the actual wording of the problem (and any instructions for the set of problems), so we can be sure what is expected.
OK so you don't actually have to find the sum you just need to show it is less than or equal to 1000. That makes it a bit easier. (BTW are you sure the 1000 is not a typo??)
Please let this be a lesson; if you continue to post, please always give us the complete question. It saves time.
Now I see that they want you to use the 5th property of summation in the page you showed! Since you didn't mention an inequality, I didn't consider that as being relevant.