Rules for summations

[Halo]

Hello

 

These notes were taken from a lecture I had the other day:

 

[math]\sum\limits_{k=0}^m \binom{m}{k}x^k + \sum\limits_{k=0}^m \binom{m}{k}x^{k+1}[/math]

[math]= \sum\limits_{k=0}^m \binom{m}{k}x^k + \sum\limits_{k}^{m+k} \binom{m}{k}x^{k(k+1)}[/math]

[math]= \binom{m}{0}x^0 + \sum\limits_{k=1}^m (\binom{m}{k} + \binom{m}{k-1})x^k + \binom{m}{m}x^{m+1}[/math]

[math]= 1 + \sum\limits_{k=1}^m \binom{m+1}{k}x^k + x^{n+1}[/math]

[math]= \sum\limits_{k=0}^{m+1} \binom{m+1}{k}x^k[/math]

 

Can someone please explain to me by which rules these steps are possible, because it is all greek to me at the moment. The steps are part of a proof for the Binomial Theorem.

 

Thanks in advance!

[mathematic]

The second term on the second line looks wrong. What is the summation variable? What are the limits? The exponent k(k+1) is wrong.