How can I prove that $\sum_{n=1}^\infty \dfrac{n^4}{2^n} = 150 $? I can easily prove that this series converges, but I can't imagine a way to prove the statement above. I tried with the techniques for finding the sum of $\sum_{n=1}^\infty \frac{n}{2^n}$, but I didn't find an useful connection with this case. Could someone help me?

Consider
$$S=\sum_{n=1}^\infty n^4 x^n$$ and now, use the trick
$$n^4=n(n-1)(n-2)(n-3)+6 n(n-1)(n-2)+7 n(n-1)+n$$ which makes
$$S=x^4\left(\sum_{n=1}^\infty x^n\right)''''+6x^3\left(\sum_{n=1}^\infty x^n\right)'''+7x^2\left(\sum_{n=1}^\infty x^n\right)''+x \left(\sum_{n=1}^\infty x^n\right)'$$
