Help with testing the convergence of a series

[bolzano]

Hi i have to show that the series 1+2r+r²+2r³+r⁴+2r⁵+... converges for r=$\frac{2}{3}$ and diverges for r=$\frac{4}{3}$ using the n^th root test.

The sequence $\sqrt[n]{a_{n}}$comes a bit complicated so i was wondering if I can remove the 1st term a₁=1 and show that 2r+r²+2r³+r⁴+2r⁵+... converges, using the test of course.

Am i doing something wrong?

Thanks :)

 

[HallsofIvy]

Yes you can remove any finite number of terms and not change whether the series converges or not.

 

[bolzano]

So $(\sqrt[n]{a_{n}})$$=2r,r,\sqrt[3]{2}r,r$,$\sqrt[5]{2}r,...$ which converges to $r$ right? :)