Identify the common ratio of the geometric series.

A geometric series $\sum_{r=1}^{\infty} ar^{r-1}$ converges if and only if the common ratio $r$ satisfies $|r| < 1$.
For the given series $\sum_{r=1}^{\infty} \left(\frac{3}{x+2}\right)^r$, the common ratio is $r = \frac{3}{x+2}$.

This inequality can be rewritten as $3 < |x+2|$.
This implies two separate inequalities:

For the first inequality:

For the second inequality:

The range of values for $x$ for which the series converges is $x < -5$ or $x > 1$.

The range of values for $x$ is $\boxed{x < -5 \text{ or } x > 1}$.

3b) The sum of the first $n$ terms of a series is $S_n = 5(2^n - 1)$. Find the general term ($a_n$) and show that the sequence is a G.P.

The general term $a_n$ is given by $a_n = S_n - S_{n-1}$ for $n > 1$, and $a_1 = S_1$.

First, find $a_1$:

So, $a_1 = 5$.

Next, find $S_{n-1}$:

Now, find $a_n$ for $n > 1$:

This formula also holds for $n=1$ ($a_1 = 5 \cdot 2^0 = 5$).

The general term is $\boxed{a_n = 5 \cdot 2^{n-1}}$.

A sequence is a G.P. if the ratio of consecutive terms is constant. This constant ratio is the common ratio, $r$.
We need to show that $\frac{a_n}{a_{n-1}}$ is a constant for all $n > 1$.

Using the general term $a_n = 5 \cdot 2^{n-1}$:

Now, calculate the ratio:

Since the ratio of consecutive terms is a constant value of 2, the sequence is a Geometric Progression.