Simplifying Expressions involving Sigma

Question

After reviweing the solutions to a question involving the Biomial Theorem, I arrived at a step, where i was unsure how it occured. Specifically, i was confused about the logic of:

• k=0 -> k=1
• n-1 -> n,

and so on...

I was wondering:

1. What is the maths behind this step (at the bottom)
2. Is there a general rule for simplifying / manipulating variables involving Sigma in this way?

The question was: Use the Binomial Theorem to derive the equation:

$$n(1+x)^{n-1}=\sum_{k=1}^n C(n, k) k x^{k-1}, x \in \mathbb{R}$$

I will only share the step that I was unsure about below, the full solution will be attached for clarity 1.

Step:

Now multiply by $n$ to obtain $$ n(1+x)^{n-1}=n \sum_{k=0}^{n-1} C(n-1, k) x^k=n \sum_{k=1}^n C(n-1, k-1) x^{k-1} $$

Answer 1

It's re-indexing. $\sum_{k=1}^5 f(k) = f(1) + f(2) + f(3) + f(4) + f(5)$, and $\sum_{k=0}^4 f(k+1)$ is also $f(1) + f(2) + f(3) + f(4) + f(5)$. Therefore, $\sum_{k=1}^5 f(k) = \sum_{k=0}^4 f(k+1)$.

In your example, $\sum_{k=0}^{n-1} C(n-1, k) x^k = C(n-1, 0) x^0 + \ldots + C(n-1, n-1) x^{n-1}$, which is equal to $\sum_{k=1}^n C(n-1 , k-1) x^{k-1}$ as the second sum is also $C(n-1, 0) x^0 + \ldots + C(n-1, n-1) x^{n-1}$.