Answer
series converges to $2$
Work Step by Step
The given series can be summed up as: $\Sigma_{n=1}^{\infty} \dfrac{1}{1+2+3+4+.....}=\Sigma_{n=1}^{\infty} \dfrac{1}{\dfrac{n(n+1)}{2}}$
We will apply the limit comparison test with $a_n=\Sigma_{n=1}^{\infty} \dfrac{1}{\dfrac{n(n+1)}{2}}$ and $b_n=\Sigma_{n=1}^{\infty} \dfrac{1}{n^2}$
We can see that the series $b_n$ shows a convergent p-series with $p=2 \gt 1$.
Next, we have $\lim\limits_{n \to \infty} \dfrac{a_n}{b_n} =\lim\limits_{n \to \infty} [\dfrac{\dfrac{1}{\dfrac{n(n+1)}{2}}}{1/n^2}]$
or, $=\lim\limits_{n \to \infty} \dfrac{2}{1+1/n}$
or, $=2$; a finite and positive term
Hence, we can see that the given series converges to $2$ by the limit comparison test.
