I'm trying to figure out what the infinite sum of $\dfrac 1{2^{n(n+1)/2}}$ is.
That's $\dfrac 12 + \dfrac 1{2^3} + \dfrac 1{2^6} + \dfrac 1{2^{10}} + \cdots $ where the powers $1,3,6,10,\cdots$ are the triangular numbers.
It's convergent as its bounded above by the sum of $\left(\dfrac 12\right)^n = 1$.
I've noticed that the ration of the terms, that is $\dfrac{a_{n+1}}{a_n} = \left(\dfrac 12\right)^n$, and so the ratios form a geometric series, but can't seem to find a way to possibly use this.
I was wondering if anyone could give me a hint or a push in the right direction? Thanks