Log-linearizing optimal price in New Keynesian model

[Charlotte87]

Homework Statement

I am going to do a log-linearization around a zero-inflation flexible price steady state of:

$\frac{P_{t}^{*}}{P_{t}}E_{t}\sum_{k=0}^{\infty}\theta^{k}\beta^{k}C_{t+k}^{1-\sigma}\left(\frac{P_{t+k}}{P_{t}}\right)^{\epsilon-1}$

Zero-inflation flexible price steady state implies that $P_{t} = P^{*}_{t} = P_{t+k} \equiv \bar{P}$.

The Attempt at a Solution

I know the solution is:
$\frac{\bar{C}^{1-\sigma}}{1-\theta\beta} +\frac{\bar{C}^{1-\sigma}}{1-\theta\beta}(\hat{p}^{*}_{t} - \hat{p}_{t}) + \bar{C}^{1-\sigma}\sum_{k=0}^{\infty}\theta^{k}\beta^{k}[(1-\sigma)E_{t}\hat{c}_{t+k} + (\epsilon-1)(E_{t}\hat{p}_{t+k} - \hat{p}_{t})]$

where $\hat{x}_{t} = x_{t} - \bar{x}, x_{t} = \log(X_{t}), \bar{x} = \log(\bar{X})$ where $\bar{X}$ is the steady state value of X. And we have used $\sum_{k=0}^{\infty}\theta^{k}\beta^{k} = \frac{1}{1-\theta\beta}$.

I've now tried for several hours to do this approximation, but I just do not get to the solution. Can anyone help me on the way?

 

[Abraham Vela]

It is straightforward. Please let me know if you have not figured this out yet, since your post is quite old.
Abe