Finding the end of an interval of convergence

Question

I am given

$$a(x) = n(6x-1)^n$$

I have calculated the ratio: $a(n+1)/a(n)$, and have found that as $n$ approaches infinity, the limit approaches $6x-1$.

I am asked to find the left hand end of this interval of convergence.

I calculated $6x-1 < 1$ (as the ratio must be less than one for convergence), to which i got $x<1/3$.

From this, I think that the range of interest would be from $-1/3$, to $1/3$, and I will need to check these endpoints for convergence too. But, the answer to this question is $0$. How can I "test" this range to find that the left-most end of the interval of convergence is $0$?

Answer 1

Your way is right, the ratio test (or the root test) leads to the condition

$$|6x-1|<1 \iff -1<6x-1<1 \iff 0<x<\frac13$$

and we have

• $a(0) = n(6x-1)^n=(-1)^nn$

• $a(1/3) = n(6x-1)^n=n$