Complex Variables: Proving 1+2w+3w^2+...+nw^(n-1) = n/(w-1)

[jjangub]

Homework Statement

Let w = e^((2pi*i)/n). Show that 1+2w+3w^2+...+nw^(n-1) = n/(w-1)

Homework Equations

1+x+x^2+x^3+...+x^m = (1-x^(m+1))/(1-x) --> A
1+x+x^2+x^3+... = 1/(1-x) --> B

The Attempt at a Solution

First of all, multiply (w-1) on both sides, then we get
w+2w^2+3w^3+...+(nw^n)-1-2w-3w^3-...-nw^(n-1) = n we simplify left side,
-1-w-w^2-w^3-...-w^n+(nw^n) = n add -1 on both sides
1+w+w^2+w^3+...+w^n-(nw^n) = -n
1+w+w^2+w^3+...+w^n = (nw^n)-n for the left side, we know from A that
(1-w^(n+1))/(1-w) = (nw^n)-n
But I can't get left side and right side equal.
Did I use the right method? Which part is wrong?
Please tell me...
Thank you

 

[mighty2000]

.

Hello, thank you for your question. Your approach is correct, but there is a small error in your simplification. Let's start from where you simplified the left side:

-1-w-w^2-w^3-...-w^n+(nw^n) = n

You correctly added -1 on both sides, but then you made a mistake in the next step. The term -nw^(n-1) should be -nw^n. So the correct simplification is:

-1-w-w^2-w^3-...-w^n+(nw^n) = n
-1-w-w^2-w^3-...-w^n-(nw^(n-1)) = n

Now, you can continue with your approach:

1+w+w^2+w^3+...+w^n-(nw^n) = -n
(1+w+w^2+w^3+...+w^n)/(1-w) = -n/(1-w)
Using B, we can simplify the left side to 1/(1-w) and we get:
1/(1-w) = -n/(1-w)
1 = -n
Which is obviously not true. So where did we go wrong?

The mistake is in the step where we simplified -nw^(n-1) to -nw^n. This is only true if n is an even number. When n is an odd number, we cannot simplify -nw^(n-1) to -nw^n. So, in order to make our approach work for both even and odd numbers, we need to consider two cases:

Case 1: n is an even number.
In this case, we can simplify -nw^(n-1) to -nw^n and continue with our approach as shown above.

Case 2: n is an odd number.
In this case, we need to use a different approach. We can start from the original equation:

1+2w+3w^2+...+nw^(n-1) = n/(w-1)

Multiply both sides by (w-1):

(1+2w+3w^2+...+nw^(n-1))(w-1) = n

Expand the left side and simplify:

w+2w^2+...+nw^(n-1)-1-2w-3w^2-...-nw^(n-1) = n
w+2