Mathematical Induction Proofs I've really been struggling with mathematical induction. I've been trying to follow my book the best I can. I feel like my proofs are correct. But then again I have my doubts. If anyone can give me some insight on weather I am doing it correctly or not that would be awesome.

I don't want these answered for me I just want some good feedback.
 A: Your base cases suffer from high-school-style mute writing. For example:

$\sum\limits_{i=1}^1 \left(i * i!\right) = \left(1+1\right)! - 1 = 1$
$1 = 1$

This should be something like

In this case, the claim we must prove is $\sum\limits_{i=1}^1 \left(i * i!\right) = \left(1+1\right)! - 1$.
But this is true, since it boils down to $1 = 1$.

Note the difference: The words explain what the formulas are for. The formulas in themselves aren't a complete argument, as you are reasoning backwards, so the first formula is not in itself obvious but rather follows from the second. (Even if you were reasoning forwards, it would help to insert a word or two.)
Critiquing some minor points:

*

*No one uses the $*$ sign for multiplication outside of ASCII-only environments. In LaTeX, use "\cdot" ($\cdot$).


*Proofs should start with "Proof." or a similar delineation. (This is, at least, standard since the 1950s or so.)


*Induction steps should start with "Induction step." or a similar delineation.


*"Since" starts a dependent clause, which cannot constitute a whole sentence by itself. Thus, "Since ... The result will be even" should be "Since ..., the result will be even".
In case of doubt, try to look through the eyes of a reader, and imagine the proofs are several times longer and there are many more of them. What changes would improve the reader's experience?
Mathematically, your proofs are correct.
