Summation notation $ 1 + \frac{1}{1+2} + \frac{1}{1+2+3} + \ldots + \frac{1}{1+2+3 + \ldots +n} + \ldots $

Question

How to write the sum
$$
1 + \frac{1}{1+2} + \frac{1}{1+2+3} + \ldots + \frac{1}{1+2+3 + \ldots +n} + \ldots
$$
in summation (∑) notation.There are 2 series here, one the entire 1 + (1/1+2)… series and the other one in denominator of each term. This summation is a sub-part of a Induction problem.

Answer 1

$$\sum_{n=1}^{\infty}\frac{2}{n(n+1)}$$ which is equivalent to $$\sum_{n=1}^{\infty}\frac{1}{T_n}$$ $T_n$ being the $n^{th}$ triangle number.

Clearly, it converges to $2$. This is because the sequence is equivalent to

$$\sum_{n=1}^{\infty}\frac{2}{n}-\frac{2}{n+1}$$ which is telescoping.