value of $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+\cdots$? Is there anything known about the value of the series $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+\frac{1}{1+2+3+4+5+6}+\cdots$ ?

The denominators $a_n$ are
$$
a_n = \sum_{k=1}^{n}k = \frac{n(n+1)}{2}
$$
so
$$
\sum_{n=1}^{\infty} \frac1{a_n} =
\sum_{n=1}^{\infty} \frac{2}{n(n+1)} =
2\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right) = 2
$$
If you don't have the resources to check this on a computer (out to infinity),
you will have your computer or calculator carry out the calculation for $n<N$.
You should then get a number near the partial sum
$$
s_N=\sum_{n=1}^{N-1} \frac1{a_n} = 2 \left( 1 - \frac{1}{N} \right),
$$
where whatever difference you find between the computer's answer and this expected answer will be the result of truncation or rounding errors due to the machine representation of each floating point number (which usually involve binary mantissas as in IEEE formats) encountered along the way.
