summation of $\sum_{n=1}^\infty \frac {2^n+4^n}{6^n}$ So I need to show that this sum converges to 4.5. But when i did this is got the sum converges to 2.5.
$$\sum_{n=1}^\infty \frac {2^n+4^n}{6^n}$$
My workings:
$$\sum_{n=1}^\infty \frac {2^n+4^n}{6^n}=\sum_{n=1}^\infty \frac {2^n}{6^n}+\sum_{n=1}^\infty \frac {4^n}{6^n}$$ 
The two summations are then two converging geometric series, whihc the first is 0.5 and the second is 2 so the overall sum converges to 2.5 - but the sum converges to 4.5. Any help would be great.

Your result seems correct indeed recall that for $|r|<1$
$$\sum_{k=0}^\infty r^n = \frac1{1-r} \implies \sum_{k=1}^\infty r^n = \frac1{1-r} -1$$
therefore
$$\sum_{n=1}^\infty \frac {2^n+4^n}{6^n}=\sum_{n=1}^\infty \left(\frac13\right)^n+\sum_{n=1}^\infty \left(\frac23\right)^n=\frac12+2=\frac52$$
while
$$\sum_{n=0}^\infty \frac {2^n+4^n}{6^n}=\sum_{n=0}^\infty \left(\frac13\right)^n+\sum_{n=0}^\infty \left(\frac23\right)^n=\frac32+3=\frac92$$
