# Finding the sum of an infinite series

1. Dec 4, 2009

### szklany

Find the sum of the infinite series

$$\sum _{n=1}^{\infty } \left( i /2\right) ^{2\,n}$$

I just can't seem to get started on this problem, so I was hoping somebody could give me a hint, as to what methods i should read up on.

2. Dec 4, 2009

### clamtrox

A good place to start would be to consider what (i/2)^2 is, that simplifies the sum considerably. You also need the formula for geometric series, that you can find for example from wikipedia.

3. Dec 5, 2009

### r.a.c.

Actually, break i into its polar form which is $$i^2^n = e^i^n^(^p^i^)$$ . Now expand using Euler's identity and you are left only with $$cos\ n(pi) = (-1)^n$$ From there it is a geometric series.

4. Dec 5, 2009

### HallsofIvy

Staff Emeritus
Actually, there is no need to worry about "i". $(i/2)^{2n}= (i^2/4)^n= (-1/4)^n$ so this is a purely real geometric series.

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