## Neat Number Theory/Discrete Math Problem

**Point of post: **In this post I will discuss the following simple problem:

” Let , then define . Compute “

where stands for . We will solve this problem in two ways, the first of which is simple but the second of which generalizes to a much, much more general case which we will discuss in the next post.

**Problem:** Let , then define . Compute .

*Initial Proofs*

**Proof (1): **Note first that converges. To see this we first note that is increasing since

Next we recall the well known fact that

So, in particular thus . So, and so by induction and since we may conclude that . Thus, if is defined as reduction we can see that

note though that since we see that

are integers and so applying our above observation about we note that

and

so we may conclude that

and since . Thus, we may conclude by the Squeeze Theorem that

**Proof (2): **In this post we find explicitly the value of and from there conclude. We begin by noting that

thus, it clearly follows that if

and

that

(where is merely notation for and , i.e. that forms a partition of ). Thus,

so that

thus it suffices to compute . To do this we recall the simple law that

Now, it is fairly obvious that

and

and since

it follows that

Putting this all together we get that

so that

But, evidently since in general

we may conclude by the Squeeze Theorem that

.

[…] Point of post: This is a continuation of this post. […]

Pingback by Neat Number Theory/Discrete Math Problem (Pt. II) « Abstract Nonsense | November 4, 2010 |

[…] for some odd reason, I thought I’d also generalize the methodology in the first proof of this post to the general form and give another alternate proof. These methods have both their pros and cons. […]

Pingback by Neat Number Theory/Discrete Math Problem (Pt. III) « Abstract Nonsense | November 6, 2010 |