Strange manipulation with a dummy variable to solve a summation I get sometimes summations where I need to change a number by a dummy variable to integrate/differentiate the sum to get the closed form.
The method works but it seems too strange changing a parameter by a variable to manipulate something and after revert the change to get the solution. An example:
$$\sum_{k\ge 0}k3^k\binom{n}{k}\overset{\text{strange temporal manipulation}}{\longrightarrow}\sum_{k\ge 0}kx^k\binom{n}{k}$$
Then
$$\sum_{k\ge 0}kx^k\binom{n}{k}=x\sum_{k\ge 0}kx^{k-1}\binom{n}{k}=x\frac{\mathrm d}{\mathrm dx}\left(C+\sum_{k\ge 0}x^k\binom{n}{k}\right)=\\=x\frac{\mathrm d}{\mathrm dx}\left(C+(x+1)^n\right)=nx(x+1)^{n-1}$$
Now I change back to get the solution (what is correct)
$$\sum_{k\ge 0}k3^k\binom{n}{k}=3n4^{n-1}$$
Two questions:


*

*We need something special to justify these manipulations?

*There are better methods to solve these sums without the need of this manipulation? Im searching specifically, if exist, some alternative integrate/differentiate that dont need this kind of dummy variable method
Thank you in advance.

Note that $k{n\choose k}=n{n-1\choose k-1}$. So $$\sum_{k\ge0}k3^k{n\choose k}=\sum_{k\ge0}n3^k{n-1\choose k-1}=3n\sum_{k\ge0}3^{k-1}{n-1\choose k-1}=3n4^{n-1}$$
