Question

Identify whether these series are divergent or convergent geometric series and find the sum, if possible.

Answer 1

Well, first, a geometric series \( \sum_{k = 0}^\infty r^k \) converges only if \(|r| < 1\)
In the first case, we can say:
$$
\sum_{i=1}^\infty 12\left( \frac{3}{5} \right) ^{i-1} = 12 \cdot \sum_{i=1}^\infty \left( \frac{3}{5} \right) ^{i-1} = 12 \cdot \sum_{k=0}^\infty \left( \frac{3}{5} \right) ^{k}
$$And get it to the from above. and we can clearly see that \(r = \frac{3}{5}\) and therefore the series converges. If the series converges it has a sum.
The only option matches is B, but we can still find the sum:
$$
S = \sum_{i=1}^\infty \left( \frac{3}{5} \right) ^{i-1} = \left( \frac{3}{5} \right)^0 + \left( \frac{3}{5} \right)^1 + \left( \frac{3}{5} \right)^2 \dots \\
\frac{3}{5} \cdot S = \left( \frac{3}{5} \right)^1 + \left( \frac{3}{5} \right)^2 + \left( \frac{3}{5} \right)^3 \dots \\

\begin{align*}
S - \frac{3}{5} \cdot S = \left( \frac{3}{5} \right)^0 + & \cancel{\left( \frac{3}{5} \right)^1 + \left( \frac{3}{5} \right)^2 + \left( \frac{3}{5} \right)^3 \dots} \\
- \Bigg[ &\cancel{ \left( \frac{3}{5} \right)^1 + \left( \frac{3}{5} \right)^2 + \left( \frac{3}{5} \right)^3 \dots }\Bigg] = \left(\frac{3}{5} \right)^0 = 1
\end{align*} \\
\left(1 - \frac{3}{5} \right)S = 1 \\
\frac{2}{5} \cdot S = 1 \\
S = \frac{5}{2} \\
12 \cdot S = 12 \cdot \frac{5}{2} = 6 \cdot 5 = 30
$$This technique is used at wikipedia in the page of geometric series:
http://en.wikipedia.org/wiki/Geometric_series#Example
and acquires the sum for convergent series. divergent series have no sum and such techniques will yield weird results. For example:
https://www.youtube.com/watch?v=PCu_BNNI5x4
Of course those people are youtubers so their hobby is blow minds, but divergent series simply do not have a sum. the second question:
$$
\sum_{i=1}^\infty 15\left( 4 \right) ^{i-1} = 15 \cdot \sum_{i=1}^\infty \left( 4 \right) ^{i-1}
$$ has \(r = 4\) and therefore is divergent, so it has no sum.

Answer 2

A better example of weird answers with such techniques on divergent series:
https://www.youtube.com/watch?v=w-I6XTVZXww
It is actually some kind of part 2 of the video above