- #1
opus
Gold Member
- 717
- 131
Homework Statement
Express the sum as a fraction of whole numbers in lowest terms:
##\frac{1}{1⋅2}+\frac{1}{2⋅3}+\frac{1}{3⋅4}+...+\frac{1}{n(n+1)}##
Homework Equations
The Attempt at a Solution
Please see attached image for my work. The reason I am posting the image rather than typing this out is because I don't really have a reason, but I do have the solution and the picture shows how I came to the solution. Basically what I did was to see any kind of general pattern to get a general idea of what the solution was, then try to develop mathematical reasoning that backed up what I found as the potential solution.
As you can see, my conclusion was that the solution is ##\frac{n}{n+1}##, which is verified by the back of the book. However, I don't like the way that I got the solution. This particular question was in a chapter of partial fraction decomposition, so I believe that a more thorough argument could be made by using that, but I don't see how. Any ideas?