Determine con-/divergence of $\sum\limits_{n=1}^{\infty}\frac{1+2+\cdots+n}{2^n}$ **Problem:** Determine if $\sum\limits_{n=1}^{\infty}\frac{1+2+\cdots+n}{2^n}$ converges or diverges. **My attempt:** I'm having a hard time with this one. * Trying the ratio test, I'm unable to simplify the expression * Trying the root test, I get the nth root of a sum, which I don't know how to simplify * Trying the integral test, I'm having a hard time performing the actual integration * Trying the (limit) comparison test, I can't find any reasonable expression to compare it with, but that might be due to the fact that I've been working on sequences like this for less than a day Any help appreciated!

Notice that $1+\dots+n\leqslant n^2$, then you can conclude convergence with the ratio test.