I hope this is the right forum for this, it is a differential equations I'm doing, with series now which I'm reviewing before getting into the calculus of it.
I have to find the radius and interval of convergence for this series, and i'm using a ratio test but I can't get past a certain point. my original equation is:
I then applied the ratio test (i.e. n+1 series on top and original series on the bottom) and get this far:
Seeing as its the harmonic series whose sum is infinity i am baffled at how to simplify this so i can continue with my equation to get my radius and interval of conversion.
Any advice really appreciated
Thanks guys, was my calculation of the ratio correct though ? because the limit for 1/n as n tends to infinity works out to 0. So I can conclude that there is no radius of convergence as the series is divergent?
You did not apply the ratio test correctly for several reasons. First, because the ratio test only applies to series of positive numbers. That is why girdav has the absolute value in . You should not have that "-".
Also, you have done the cancelation wrong. but . Now the ratio is
Of course, that still goes to 0 so the "radius of convergence" is infinite- this series converges for all x.
In fact, it is just the MacLaurin series for cos(x).