<s>[INST] Summation notation $ 1 + \frac{1}{1+2} + \frac{1}{1+2+3} + \ldots + \frac{1}{1+2+3 + \ldots +n} + \ldots $ 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.
 [/INST] $$\sum_{n=1}^{\infty}\frac{2}{n(n+1)}$$ 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.
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