Is $\sum_{n=1}^\infty nc_n(x-1)^n$ equivalent to $\sum_{n=0}^\infty nc_n(x-1)^n$?

Question

Are these expressions equivalent?

$$ \sum_{n=1}^\infty nc_n(x-1)^n \qquad\text{and}\qquad \sum_{n=0}^\infty nc_n(x-1)^n $$

My reasoning is that they both start at zero the first goes the following pattern:

Pattern 1 $$\sum_n^\infty a_n= c_1(x-1)^1+2c_2(x-1)^2\dots$$

Pattern 2 $$\sum_n^\infty b_n= 0+ c_1(x-1)^1+2c_2(x-1)^2\dots$$

Please let me know if my logic is sound please? If they aren't could you please explain?

I added this differential equation tag because it belongs to a DE I am manipulating the indexes for.

Answer 1

Yes, you are correct.

Just be cautious that $c_n$ should be a term that you expect that it can be evaluated with value $n=0$, say $c_n=\frac1n$ shouldn't be there.