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I have a problem with computing this geometric series.

**1. Homework Statement**

I have to compute

[tex]\sum_{i=0}^\infty{(\frac{1}{2z})^{2k}} + \sum_{i=0}^\infty{(\frac{1}{3z})^{2k+1}}[/tex].

It's for computing the z-transform of

[tex]f[k]=0[/tex] for [tex]k<0[/tex]

[tex]f[k]=(\frac{1}{2})^k[/tex] for [tex]k=0,2,4,6,...[/tex]

[tex]f[k]=(\frac{1}{3})^k[/tex] for [tex]k=1,3,5,...[/tex]

**2. Homework Equations**

**3. The Attempt at a Solution**

It's the [tex]2k[/tex] and [tex]2k+1[/tex] that annoys me in the sum.

I tried

[tex]\sum_{i=0}^\infty{(\frac{1}{2z})^{2k}}=\sum_{i=0}^\infty{(\frac{1}{4z^2})^{k}}[/tex]

but I don't know if that helps?

Thanks,

Yoran

**1. Homework Statement**

**2. Homework Equations**

**3. The Attempt at a Solution**