1.

Note the obvious, the only thing that will dominate this series so that it converges is a geometric term. Therefore

The Root Ratio test will give

2.

Apply Root test. First noting that

and the continuity of the exponential function. The limit tends to x.

After noting what I said above we just have to note that

So

3. Note that

Which converges for all x.

Or hey just use the Root test if its not obvious

4. I don't see how this is differnt than number three?

Either note that

And since the right is the power series for

which is convergent for all x the left side is as well (noting that the left side is strictly greater than zero)

Or just use the root test

5. Using the Root test so we have that

I leave the endpoints up to you.