Please do these series diverge or converge and find their sums if they converge

Question

a) $$\sum_{k=1}^\infty 3^{-2k+1}$$ b) $$\sum_{k=1}^\infty 3^{2k+1}$$

my Trial

a) $$\sum_{k=1}^\infty 3^{-2k+1} = \sum_{k=1}^\infty \frac{-3}{9^k}$$

I am blocked because I wanted to have it in the form $ar^n$ and later us

Answer 1

For a), $r=1/9$; for b), $r=9$. So b) diverges, but... well, I'll let you work out a).

Answer 2

Note that

• $\sum_{k=1}^\infty 3^{-2k+1}=3\sum_{k=1}^\infty \left(\frac19\right)^k$

and

• $\sum_{k=1}^\infty 3^{2k+1}=3\sum_{k=1}^\infty 9^k$