Define a function

where denotes the greatest integer less than or equal to , also called the floor function.

Then, define a function

- Draw the graph of for and prove that is periodic with period 1.
- For , prove
and prove that is also periodic with period 1.

- Give a formula for in terms of the floor of .
- Find a such that
- For the constant from part (d), define
Prove that is periodic with period 1 and that if , we have,

- The graph of is as follows:
Then, we prove that is periodic with period 1.

*Proof.*We compute to show (where I’ve replaced Apostol’s notation with since this less likely to cause confusion, also we use the solution to this exercise in the second line) -
*Proof.*First, we establish the requested formula,This was the requested formula. Next, to show is periodic with period 1, we show for any ,

Then, using the formula in the first part of (b) we have

Further, from part (a) we know is periodic with period 1, so , and we have,

by definition of . Thus, is periodic with period

- To express in terms of of we compute as follows:
where since is an integer by definition, and has period 1, so for any integer . Continuing,

Here, we know the term in the integral is zero since for all since . Then,

This is the requested expression of in terms of .

- Here we compute the integral, using the formula for we established in part (c), and solve for the requested constant ,
- First we give the proof that is periodic with period 1.

*Proof.*We compute,Next, we establish the requested formula. If , we have

as requested.

Hi, can someone help me understand why Q(1)=0?

it seems like it should be Q(1)= c , and Q(x+1)=Q(x)+c

in 1.e ,

Thanks

got it..never mind