How do I change the variable in this expression?

[PolyFX]

Homework Statement

Rewrite the following expression as a single product.
Hint: Perform a change of variable first.

Homework Equations

The Attempt at a Solution

I looked at the example from the book regarding a change of variable. They first started out by calculating the upper and lower limits of the expression. However, in this the upper limits and lower limits are different. For example, the right product expression has an upper limit of n - 1 while the left has an upper limit of n. Furthermore, I can make for example j = k+1 but then what about k+2? Would I need to introduce a new variable or do i make j = (k+1) + 1 to represent k +2?

How do I go about performing a change of variable with this expression?

 

[tiny-tim]

… Would I need to introduce a new variable or do i make j = (k+1) + 1 to represent k +2?

How do I go about performing a change of variable with this expression?

HI PolyFX!

You'll avoid mistakes if you do introduce a new variable, to go from 0 to n-1.

You can then turn that new variable back to k (it's only a "dummy" variable ) in the next step.

 

[PolyFX]

Hi sorry for the late reply,

I am still somewhat confused about how to approach this question. So far this is how I've startend it.

Let j = K+1/K+2

Therefore when k = 0,
j = 1/2
and when K = n - 1,
j = (n-1) + 1 / (n-1 ) + 2
j = n


So I introduce another variable for the other product expression?

so let x = K+1/K+2
when k = 1
j=2/3
and when k = n
j = n+1/n+2

I am stuck here.


To me I get the feeling that I am definitely doing something wrong. Would be great if someone get further help me out here.


-Thank you

 

[HallsofIvy]

Hi sorry for the late reply,

I am still somewhat confused about how to approach this question. So far this is how I've startend it.

Let j = K+1/K+2

No! You've completely misunderstood. The point was to change the "variable" (index), not the expression itself!

The first product goes from k=0 to n-1. If we let j= k+1, it will go from j= 0+1= 0 to (n-1)+ 1= n, just like the second product.
Now, change the expression to j: since j= k+1, k= j- 1 and so k+ 1 becomes j while k+2 becomes j-1+ 2= j+1. Now you have
$$\left[\Pi_{j=1}^n\frac{j}{j+1}\right]\left[\Pi_{k=1}^n\frac{k+1}{k+2}\right]$$

Now, since the "k"s are dummy indexes, you can just let the "j" in the first product be "k" to get
$$\left[\Pi_{k=1}^n\frac{k}{k+1}\right]\left[\Pi_{k=1}^n\frac{k+1}{k+2}\right]$$
and now you can multiply those.

Therefore when k = 0,
j = 1/2
and when K = n - 1,
j = (n-1) + 1 / (n-1 ) + 2
j = n


So I introduce another variable for the other product expression?

so let x = K+1/K+2
when k = 1
j=2/3
and when k = n
j = n+1/n+2

I am stuck here.


To me I get the feeling that I am definitely doing something wrong. Would be great if someone get further help me out here.


-Thank you