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

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.

• Your logic is correct – Ninad Munshi May 15 at 14:42

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.