Very elementary math question here in regards to my discrete math class. I've got a problem here that says...
"Compute the following":
$$\sum_{j=1}^n \frac{1}{j(j+1)}$$
What on earth does it mean to "compute" this? I mean, without any values, does that just mean to find another way of re-arranging the components to get a formula of equal value, just in a different form?
I did some iterations by hand and found that for n=4, we end up with:
$$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}$$
So it looks like each iteration just increases the denominator by the previous increase's next even number:
$\frac{1}{2} \rightarrow \frac{1}{6}$ (increase by 4)
$\frac{1}{6} \rightarrow \frac{1}{12}$ (increase by 6)
$\frac{1}{12} \rightarrow \frac{1}{20}$ (increase by 8)
So I see the pattern. Does that mean for example that I could re-write the above sigma summation as a "rule" for making the sequence?