Table 1.1:
The same results of from Table 1.1 can be represented as follows:
Table 1.2 :
From the patterns exhibited in Table 1.2, I notice that which is further illustrated in Graph 1.1.
In Graph 1.1, I plotted the graph of for the first three values (represented by green dots) and I assumed that (n+1)! will lead to a conjecture forand plotted its values for n=1,2,3 (represented by red dots) . From the two graphs, I notice that (n+1)! is exactly 1 unit above for all three points I conclude that: = (n+1)! 1
Graph 1.1
Part 4:
The conjecture that I derived in Part 3 for can be proven through Mathematical Induction:
 If n=1
LHS:! =1
RHS: (1=1
RHS=LHS=
is true

If is true, then
If k=k+1, then
Now,
=
=
=
=(k+2)!1 (k+2)!
Thus is true whenever is true and is true.
is true for all n
Part 5:
!, I use this formula to show that is also true by simplifying to equal !
! = (n+1)!n!
= (n+1)n!n! (n+1)! = (n+1)n!
= n! (n+11)
= !
!
Now, I use to device a direct proof for the expression of that I conjectured in Part 3.
!
From Part 3, I know that
When the first value of the first term is subtracted from the second value of the following term, 0 is derived so I cancel these terms. After I cancel the values to the most simplified manner, 1! and (n+1)! are left in the expression from which the following equation is derived:
The conjecture of is proven
Part 6:
From Part 5, I know that:
I use the equation of to derive an expression for by substituting n+1 for n and simplify it:
If n=n+1, then
=
Let
To express in factorial notation, I substitute and with their equivalent factorial notation forms and simplify them:
Part 7:
Let
If n=1
were
If n=2
were , !
If n=3
were , !,
Part 8:
From Part 7:
To conjecture an expression of , I first organize the results that are derived in Part 7 to discover a pattern in the value of as n varies.
Table 2.1:
The same results of from Table 2.1 can be represented as follows:
Table 2.2:
Thus, from the pattern exhibited in Table 2.2, I notice that which is further illustrated by Graph 2.1.
In Graph 2.1, I plotted the graph of for the first three values (represented by blue dots) and I assumed that (n+1)!(n+3) will lead to a conjecture for formula forand plotted its values for n=1,2,3 (represented by green dots). From the two graphs, I notice that (n+1)!(n+3) is exactly greater by 3 units to for all three points I conclude that: = (n+1)!(n+3) 3
Graph 2.1
Part 9:
The conjecture that I derived in Part 7 for can be proven through Mathematical Induction:
is:
+ (2+2)!2! + … +
 If n=1
LHS:
= (1+2)!1!
= 5
RHS: (1+1)!(1+3)= 5
is true

If is true, then
+ (2+2)!2! + … +
If k=k+1, then
Now,
=
=
= (k+1)!(k+2)
= (k+2)!(k+4)
Thus is true whenever is true and is true.
is true for all n
Conclusion:
Through this investigation, I have developed my knowledge about series and sequences involving permutations. I have learnt to use the patterns in a series to conjecture an expression for it and I had an opportunity to utilize my awareness of mathematical induction into proving the general term for the series. Most importantly, I have learnt to use technology related to series involving permutations. I enjoyed this investigation.