The series depicted in your image is:

$-\frac{2}{2} + \frac{6}{5} + \frac{10}{10} + \frac{14}{15} + \dots$

Let's break it down and analyze its general pattern.

Observations:

1. The numerator forms an arithmetic sequence: $2, 6, 10, 14, \dots$, increasing by 4 each time.
   • General formula for the numerator: $a_n = 2 + 4(n - 1) = 4n - 2$.
2. The denominator forms an arithmetic sequence: $2, 5, 10, 15, \dots$, increasing by $3, 5, \dots$ each time.
   • General formula for the denominator: $d_n = n(n + 1) / 2$, which corresponds to triangular numbers.

General Term of the Series:

The $n$-th term of the series can be expressed as: $a_n = \frac{4n - 2}{n(n + 1)/2}.$

Simplify: $a_n = \frac{4n - 2}{\frac{n(n + 1)}{2}} = \frac{2(4n - 2)}{n(n + 1)} = \frac{8n - 4}{n(n + 1)}.$

Further simplify: $a_n = \frac{8n - 4}{n(n + 1)} = \frac{8}{n + 1} - \frac{4}{n}.$

Thus, the series can be written as a telescoping sum: $\sum_{n=1}^\infty \left( \frac{8}{n + 1} - \frac{4}{n} \right).$

Would you like me to compute the sum or investigate its convergence?