Hey, I'm supposed to find out the interval of convergence for this power series:
Any thoughts?
First of all thus maybe you meanOriginally Posted by CowPizza
Well, I think it would be easier to consider,
, which is equivalent to the one you gave.
Then let , thus,
.
Notice that this is a power series, thus use the generalized ratio test, where .
Thus,
, thus the radius of convergence is the reciprocal of this which is also 1.
Let us now find the interval of convergence, since the series is absolutely convergent we have that , thus,
.
Notice when the series diverges because it forms a harmonic series and when the series converges conditionally because it form and alternating series. Thus,
is the interval of convergence.
Q.E.D.