You should learn Taylor's Theorem.
It is a way to approximate the remainder when you approximate a function with the Taylor polynomial.
Hey, would anyone mind railing off some very difficult Taylor/Maclaurin series problems, as well as series convergence/divergence problems, and even some limit ones.
The reason is that I am now able to do all the problems in any book I can find, but I know there are millions of them I probably can't do, and I would love to see them so I can learn, for that is my greatest passion. And I couldn't think of a place that would have more people with knowledge of these kind of problems than MHF.
And anyways, you get to compete in the MHF sport of embarrassing me
So any problems of those three subjects would be greatly appreciated, I will solve them and ask for criticism
Try this one to see whether it converges or not.
Here's one, similar to the other, you can try Gauss's test on:
Here is also a curious little tidbit:
What if we take the divergent harmonic series, and remove all terms that have a 9 in them?.
Is it then convergent?.
The first one converges.
If you are curious:
There is a neat corollary from Cauchy which says that if
is a positive decreasing sequence, then
and diverges if
Look at the relation:
Therefore, the series converges.
The second one converges if p>2 and diverges if
The third one, the harmonic, surprisingly converges.
It can be shown that if we delete from the harmonic series all terms that contain X, the resulting series in convergent.
I didnt want to start a new thread, but over all of my series exploration there are only a few that I cant find the divergence or convergence of,
If anyone could help
depending on the convergence of the first one this could just be a simple convergent-divergent=divergent one
Find all and
such that the following series is convergent
Those are the pesky ones ...
I am sure they are very simple, but I just dont see them, could someone enlighten me
Also any more questions woudl be appreciated
EDIT: for the first one I think you have to show that
But I am not sure