<s>[INST] Using generating functions find the sum $1^3 + 2^3 + 3^3 +\dotsb+ n^3$ I am quite new to generating functions concept and I am really finding it difficult to know how to approach problems like this. I need to find the sum of $1^3 + 2^3 + 3^3 +\dotsb+ n^3$ using generating functions. How do I proceed about it?
 [/INST] Yet another way is to start with:
$$
\sum_{0 \le k \le n} z^k = \frac{1 - z^{n + 1}}{1 - z}
$$
differentiate thrice:
$$
\frac{\mathrm{d}}{\mathrm{d} z}
   \left( z \frac{\mathrm{d}}{\mathrm{d} z}       
     \left(
       z \frac{\mathrm{d}}{\mathrm{d} z} \frac{1 - z^{n + 1}}{1 - z}
     \right)
   \right)
= \frac{1 + 4 z
          + z^2 
          - z^n (1 - 3 n^2 - n^3)
          - z^{n + 1} (4 - 6 n^2 - 3 n^3)
          - z^{n + 2} (1 - 3 n + 3 n^2 + 3 n^3)
          + z^{n + 3} n^3}
       {(1 - z)^4}
$$
Taking the limit as $z \to 1$ of this mess (l'Hôpital to the rescue) again gives:
$$
\sum_{0 \le k \le n} k^3 = \frac{n^2 (n + 1)^2}{4}
$$
The above courtesy of my CAS, maxima. Any transcription errors are mine only.
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