So, simply, where do I go wrong?

The question:

The interval of the power series is . Find .

My solution:

For the series to be convergent, d'Alemberts ratio test must be satisfied, i.e. must hold true.

For the series given we have:

.

This gives

It is easy to show that the series is convergent also for and (by making these substitutions into the series given).

So now we have that for the series to be convergent.

But the interval of convergence is , meaning that .

.

However, the key in my book claims the answer to be

Thank you in advance

Aliquantus