I must look pathetic, but I'm beginning with infinite series.
I invented to myself a very easy infinite series that I must determine its convergence. (I'm sure it converges even if I'm not able to prove it!).
The series is : .
I tried the ratio test, writing down and , dividing by I finally got . Taking the limit when tends to I got . As , the series diverges. Where is my mistake? Also, I would like to know if I can know to what number it converges.
Also
There
Thus convergent by comparison test
Also
since for a finite value of i and every term is positive
thus the integral converges by the integral test
if you would like I can give you some practice problems and help you through them
This is equal to
evaluating at 1/3 gives the correct result...which was 3/4
I just realized I made the ratio test but inversed! So I got instead of . I also notice that the series value is not equal to the result of the ratio test (yeah, why would it be so?).
From Mathstud :. Is this the value of the series? If yes, any series can be written as an improper integral and the value of the series is always equals to the improper integral?Also
since for a finite value of i and every term is positive
ToI think it's a good idea, but I'm not sure I will do a lot of them these days (since I have an exam coming very soon and it has almost nothing to see with infinite series). But why not doing a bit? So yes.if you would like I can give you some practice problems and help you through them
Finally I didn't understand well the geometric series formula. I will investigate about it.
where we want (we'll assume will be on the interval of convergence, that is )
Now the condition can be stated as:
where
Multiplying by on both sides:
That is:
Now sum from to :
Thus we have:
Thus, since :
and
You can use this method to work with other sequences where you have a recurrence