<s>[INST] Summation of more than one series How can I find $1+3x+6x^2+10x^3+.... $ for $x=6/7$. I have seen that this can be written as a sum of three series $\sum_0^\infty [(n^2/2)x^n+(3/2)nx^n+x^n]$. But I do not know how to proceed further.
 [/INST] Note that you have sum of terms of type $\frac{(n+1)(n+2)}{2}x^n$, so your sum would be:
$$1+(1+2)x+(1+2+3)x^2+(1+2+3+4)x^3+...=\\=(1+x+x^2+x^3+...)+2(x+x^2+x^3+...)+3(x^2+x^3+x^4+...)+...=\\=(1+x+x^2+x^3+...)+2x(1+x+x^2+x^3...)+3x^2(1+x+x^2+x^3+...)+...$$ Since $x=6/7$ we have $1+x+x^2+x^3+...=7$, so the above expression is: 
$$=7(1+2x+3x^2+4x^3+...)$$ Now, calculate $1+2x+3x^2+4x^3+...$ by similar method, and you will obtain result.
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