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- Homework Statement
- Prove that for every complex number ## z \neq 1 ## it occurs that ## \sum_{n=0}^{N} z^{n}=\frac{1-z^{N+1}}{1-z} ##

- Relevant Equations
- ## z= a+ib , z = r e^{i \theta} ##

I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## .

I need to show that this is equal to ## \frac{1- z^2 }{1-z} ## , i.e. ## 1+z = \frac{1- z^2 }{1-z} ##.

So I went straight ahead and did as follows:

## \frac{ 1-(a+ib)^2 }{1-(a+ib) } = \frac{ 1-(a+ib)^2 }{1-a - ib) } \cdot \frac{1- a + ib}{ 1- a + ib } = \frac{ a^{3}+i a^{2} b-a^{2}+a b^{2}-2 i a b-a+i b^{3}+b^{2}+i b+1 }{a^{2}-2 a+b^{2}+1} ## but I don't really know how to continue from here.

I also tried using ## z = r e^{i \theta } ##:

## \frac{ 1 - r^2 e^{2 i \theta } }{ 1- r e^{i \theta } } = \frac{ 1 - r^2 e^{2 i \theta } }{ 1- r e^{i \theta } } \cdot \frac{ 1- r e^{-i \theta } }{1- r e^{-i \theta }} = \frac{r^{3} e^{i \theta}-r^{2} e^{2 i \theta}-r e^{-i \theta}+1}{r^{2}-2 r \cos (\theta)+1} ## and here I also stopped, unclear how to continue.

Can you please help? I don't know how to show the base case.

Edit: Problem's solved!

I need to show that this is equal to ## \frac{1- z^2 }{1-z} ## , i.e. ## 1+z = \frac{1- z^2 }{1-z} ##.

So I went straight ahead and did as follows:

## \frac{ 1-(a+ib)^2 }{1-(a+ib) } = \frac{ 1-(a+ib)^2 }{1-a - ib) } \cdot \frac{1- a + ib}{ 1- a + ib } = \frac{ a^{3}+i a^{2} b-a^{2}+a b^{2}-2 i a b-a+i b^{3}+b^{2}+i b+1 }{a^{2}-2 a+b^{2}+1} ## but I don't really know how to continue from here.

I also tried using ## z = r e^{i \theta } ##:

## \frac{ 1 - r^2 e^{2 i \theta } }{ 1- r e^{i \theta } } = \frac{ 1 - r^2 e^{2 i \theta } }{ 1- r e^{i \theta } } \cdot \frac{ 1- r e^{-i \theta } }{1- r e^{-i \theta }} = \frac{r^{3} e^{i \theta}-r^{2} e^{2 i \theta}-r e^{-i \theta}+1}{r^{2}-2 r \cos (\theta)+1} ## and here I also stopped, unclear how to continue.

Can you please help? I don't know how to show the base case.

Edit: Problem's solved!

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