Perturbation Method [formulation]

Question

Recently when I am reading about perturbation method from following book:
Concrete Mathematics: A Foundation for Computer Science;
I came out with following formulation:
$S_n = \sum_{0\leq k\leq n}a_k$
Then we rewrite $S_{n+1}$ in two ways, by splitting off both its last term and its first term: $S_n + a_{n+1} = \sum_{0\leq k\leq n+1}a_k = a_0 + \sum_{1\leq k\leq n+1}a_k$
= $a_0 + \sum_{1\leq k+1\leq n+1}a_{k+1}$ [What is the logic behind this part of transformation, suddenly change the k to k+1 without any change in the range?]
= $a_0 + \sum_{0\leq k\leq n}a_{k+1}$ Can anyone explain that part I have mentioned here?

Answer 1

It looks fine. By introducing \begin{equation} j = k-1, \end{equation} we can write \begin{equation} a_k = a_{j+1}, \end{equation} and the domain $1\leq k \leq n+1$ corresponds to $0 \leq j \leq n$, which is equivalent to $1 \leq j+1 \leq n+1$. Hence \begin{equation} \sum_{1 \leq k \leq n+1} a_k = \sum_{1 \leq j+1 \leq n+1} a_{j+1}. \end{equation} Now you can write the dummy index $j$ on the right hand side as $k$.

Well, in any way, the step of the question looks not necessary, and one can jump straight to the last line in your question.