Question

Evaluate (1/2!)+(2/3!)+(3/4!)+...(n/(n+1)!). Conjecture the sum if n tends to infinity.

Answer 1

With some summation manipulation, we can come up with a formula for the sum of the first n terms. Then we just need to take the limit as n goes to infinity for our answer. Note that this isnt the way they want you to solve the problem. They want you to compute a couple of sums by hand (maybe up to the first 7 terms or so) and guess what the limit might be as n goes to infinity.

Answer 2

yes, as joemath says, try working out some partial sums and you may be able to spot the pattern.

Answer 3

So what we have to deal with is the sum of the first n terms which is:\[\sum_{k=0}^n\frac{k}{(k+1)!}\]I suggest maybe changing the k to (k+1)-1, simplifying, and see if you can guess what the formula for the sum of the first n terms is. Again, this is not required. You should be able to guess my computing a couple of sums by hand. The method I propose will actually give you the answer without guessing.