Need help with series index I would like to get some help if someone explain in particular the second part of the question.  I am a bit confused. I guess the first part will be $m=-2$ but don't know what will be the coefficient of $x^m$ ?
$$
\sum_{m=-3}^\infty (m+5)(m+1)a_m x^{m+1} = \sum_{m= ?}^\infty (\color{red}{??})\, x^{m} 
$$
 A: Hints:
The summand on the right is obtained by replacing $m$ by $m-1$ in the summand on the left, so you compensate by increasing the index start/stop values by $1$ (equivalently, you replace "$m=-3$" by "$m-1=-3$", which can be rearranged to "$m=-2$"). This part you have done correctly.
What remains is to continue the replacement in the coefficients. What do you get when you replace $m$ by $m-1$ in the coefficients on the left?
A: you can do the follow change of variable: $n=m+1$. So your sum is know
$\sum_{n=-2}^{\infty} (n+4)na_{n-1}x^n$ and now it has the form that you want.
A: I like to replace $m$ on the left by some other symbol, to avoid confusion with the $m$ pn the right.  Since it is just a summation index, we are free to do that.
$$
\sum_{n=-3}^\infty (n+5)(n+1) x^{n+1} = \sum_{m= ?}^\infty (\color{red}{??})\, x^{m} 
$$
Then we see that for the exponents to match, we need $n+1 = m$.  So interms of $n$,
$$
n = m-1
$$
And replacing $n$ by $m-1$ we get
$$
 \sum_{m-1= -3}^\infty (\color{red}{(m-1+5)(m-1+1)})\,   x^{m} =
 \sum_{m = -2}^\infty (\color{red}{(m+4)m})\,   x^{m} 
$$
