Could somebody explain me how to find the convergence and the radius of the convergence for the following power series?
Did you try the ratio test. When you do and take the limit,
I may as well finish.
Diverges if x<-2 or of x>0
Check the endpoints:
If we enter in x=-2 into our series we get , converges. (converges to )
If we enter in x=0, we get , diverges.
So, the interval of convergence is [-2,0) with radius of 1
My assessment agrees with MS except are you sure when x=0 it converges?.
Galactus's method is totally right, but just so you get more than one method
Root test
So we need to find all values of x such that
So we see that
Or in other words
or
Now we need to check endpoint convergence
at we have
divergence by integral test
at we have
and since
and also since
this series converges by AST
Therefoe IOC is
EDIT: For purely supplemetary purposes
Let
For the right hand limit you can use many methods, L'hopital's is most preferable here
So we have that
Use the same method to compute
Thanks so much! I fixed it before you posted that , the entire time I kept typing x instead of x+1, therefore when I saw x=0 I said a sequence of all zero terms, I am sorry for my stupid error , I knew it would bite me sooner or later
Just to show once again, now that I see it is and not
x=0 gives
and as we stated earlier the existence of eventual monotonic decrease and all the terms are positve we can apply the integral test
therefore divergent