<s>[INST] The significance of the number in front of a series? I believe I have solved the series to $-4/9$ however I'm not sure of the significance of the 9 in front of sigma. Is this simply multiplying the final result or is it something different?
The series in question is
$$9\sum_{n=1}^\infty\left(-\frac{4}{5}\right)^n$$
 [/INST] Your solution of  the  series is correct. According to the formula for geometric power series 
$$\sum_{n=1}^\infty q^n=\frac{q}{1-q}\qquad\qquad |q|<1$$
we obtain 
\begin{align*}
\sum_{n=1}^\infty\left(-\frac{4}{5}\right)^n=\frac{-\frac{4}{5}}{1+\frac{4}{5}}=-\frac{4}{9}
\end{align*}

We  do not  explicitely need to write a multiplication sign between a  factor and a series to denote the multiplication. We can simply write $9\sum_{n=1}^\infty\left(-\frac{4}{5}\right)^n$
  and the meaning is 
  \begin{align*}
9\sum_{n=1}^\infty\left(-\frac{4}{5}\right)^n=9\cdot\sum_{n=1}^\infty\left(-\frac{4}{5}\right)^n=9\cdot\left(-\frac{4}{9}\right)=-4
\end{align*}

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