I have no idea how to solve sequence problems. What is the trick? I'm
talking about convergent/divergent problems. My book does a bad job
at explaining them-it has all this jarjon like "well if X>a and n>epsilon then
for all numbers when plotted to infinity" yadda (lol I know I posted jarjon
on purpose) sorry if I sound like a jerk, I'm just frusturated at the moment.
Here's where it didn't make sense:
an=2^n/(3^n+1)
and (2n-1)!/2n+1! The author did some weird manipulation to get the solutions (trying to find out divergence or convergence) and I have no idea
what or even why he did what he did lol. Any help would be greatly appreciated!
The first step is to guess what the limit is, it seems to be .
Next we need to show can be made small.
Write, .
Now it remains to show this quantity gets small, it is easier to bound it.
Note, .
Finally note that can be made sufficiently small.
Here is how the formal proof looks like:
Given choose so that .
Then if it would imply .
Once you pick so that . Then (by induction) if it would mean .
And so if we have .
Whenever you do a delta/epsilon proof there are two parts. The mental part and the stuff you write down on paper. The mental part is what you work out on paper, that is what I did previously. What I wrote now is how it is presented on paper. The only reason why I knew how to pick such an N is because I worked it out in the other post.