Question: After the great help I received (on here), I had an easy time working with a similar progression, but this one is giving me trouble. Can anyone sum the following:

$$\begin{align}
&0.3^{-1} \\
&0.3^{-1}+0.3^{0} \\
&0.3^{-1}+0.3^{0}+0.3^{1} \\
&0.3^{-1}+0.3^{0}+0.3^{1}+0.3^{2} \\
&\cdots
\end{align}$$ and so on.

I feel very thankful for all the help I've gotten on things thus far; thank you guys, your skills are epic and regarded with great respect, by me!

Answer: 
Why do you say the integral of $\frac{1}{3x}$ not $\frac{\ln(3x)}{3}$ ? ... it is.

You say that the integral is $\frac{\ln x}{3}$, which is also right. These two integrals differ by a constant, of course.
$$\frac{\ln(3x)}3=\frac{\ln x+\ln 3}3=\frac{\ln x}3+\frac{\ln 3}3$$
This function and
$$\frac{\ln x}3$$
have the same derivative.