So I want to show that the following is true, but Iam kidna stuck... $$ \sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}\sum_{q_{3}=q_{1}}^{q_{2}}...\sum_{q_{k+1}=q_{1}}^{q_{k}}x^{q_{1}+q_{2}+...+q_{k+1}}=\sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}\sum_{q_{3}=q_{2}}^{\infty}...\sum_{q_{k+1}=q_{k}}^{\infty}x^{q_{1}+q_{2}+...+q_{k+1}} $$ At first I tought something like that: $$ \sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}x^{q_{1}+q_{2}}\prod_{m=3}^{k+1}\sum_{q_{m}=q_{1}}^{q_{m-1}}x^{q_{m}}=\sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}x^{q_{1}+q_{2}}\prod_{m=3}^{k+1}\sum_{q_{m}=q_{m-1}}^{\infty}x^{q_{m}} $$ $$ \prod_{m=3}^{k+1}(\frac{x^{q_{1}}-x^{q_{m-1}+1}}{1-x})=\prod_{m=3}^{k+1}(\frac{x^{q_{m-1}}}{1-x}) $$ $$ \prod_{m=3}^{k+1}(x^{q_{1}-q_{m-1}}-x)=1 $$ but from here on out i dont know how to continue, given that the steps so far were even productive to begin with...
In addition i dont want to use the following Indentity to prove what i have just shown. $$ \sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}\sum_{q_{3}=q_{2}}^{\infty}...\sum_{q_{k+1}=q_{k}}^{\infty}x^{q_{1}+q_{2}+...+q_{k+1}}=\prod_{m=1}^{k+1}\frac{x}{1-x^{m}}=\sum_{n=0}^{\infty}p_{k+1}(n)x^{n} $$ Thanks for all the suggestions
