Need help in binomial question and in understanding the reason for nΣk=0 changing

Question

why exactly does in this question (i and ii) in the answer sheet k=0 turn to k=1 after differentiating and also why does multiplying by (1+x^2) does the 1 expansion keep k=1 and but the x^2 expansion changes it back to k=0

sorry if this is a really easy question or something just due to the nature of my lack of knowledge, i dont even know the right terminology to even begin searching for an answer. all my friends dont know either.

Answer 1

In the following I use the notation $\binom{n}{k}:={}^{n}C_k$.

After differentiation (and division by $2$) the multiplication with $1+x^2$ gives \begin{align*} \sum_{k=0}^n k\binom{n}{k}x^{2k-1}(1+x^2)=\sum_{k=0}^n k\binom{n}{k}x^{2k-1}+\sum_{k=0}^n k\binom{n}{k}x^{2k+1} \end{align*}

Note that \begin{align*} \sum_{k=0}^n k \binom{n}{k}x^{2k-1}&=0+\sum_{k=1}^n k\binom{n}{k}x^{2k-1}=\sum_{k=1}^n k\binom{n}{k}x^{2k-1}\\ \qquad\qquad\text{and}\\ \sum_{k=0}^n k \binom{n}{k}x^{2k+1}&=0+\sum_{k=1}^n k\binom{n}{k}x^{2k-1}=\sum_{k=1}^n k\binom{n}{k}x^{2k+1} \end{align*} So, both sums can be written with index $k$ starting from $k=0$ as well as $k=1$. It's just a matter of convenience and you can choose whatever you prefer.

In fact I recommend to start both sums with the same starting value either $0$ or $1$ just to simplify reading.