Deriving formulas from Gali's book on the New Keynesian model I've found this link with all the derivations from Gali's book on New Keynesian model.
In page 13, eq. $(3.7)$, the author derives the optimal price-setting rule  $P_t^*$ by a representative firm that solves an intertemporal profit maximization problem. Equation $(3.7)$ contains two infinite series. The author then does a log-linearization on both and proceeds to the solution. Log-linearization is a local approximation, right? Then how can we be sure that the two series, with these approximations will converge?
Mathematically speaking, we are not sure. The infinite sums mentioned is of the form

$$E_t\sum_{k=0}^{\infty}\gamma^kX_{t+k},\;\; 0&lt;\gamma&lt;1$$

The fact that there is the declining factor, $\gamma^k$, provides some comfort that the sum converges, but from examples like the harmonic series we know that even if $\lim_{k\rightarrow \infty}\gamma^kE_tX_{t+k} =0$, the sum may not converge.  

Some times authors provide rigorous conditions for convergence, but I believe it is more important not to be trapped by our own tools: the use of infinite horizon is, of course, non-realistic. The only reason we use it is that it makes the mathematics more tractable. So in reality we model a problem with possibly long but finite horizon. In such a case, there is no issue of convergence. By extending the problem to the infinite horizon we silently assume that this won't destroy convergence, since it is an artificial extension.

The ultimate economic argument for convergence, is that it is meaningless to talk about economic magnitudes going to infinity. Even for prices, where it is tempting to think that they do not have an upper bound per se, we must remember that prices as a component of an economic system do have ultimate bounds.  

With prices flying to heaven, the economy goes to hell (if you excuse the christian iconography), because the price system stops functioning as a signal mechanism to co-ordinate decisions in a decentralized framework. But in the real world, when such explosions "start to happen", the society that encloses the economy in question, will intervene in one way or the other, in order to end the phenomenon. The historical experience from war-time hyperinflations and from other price-bubbles verifies this.  

Ideally, we would want a single model to encompass both possibilities and also model the possible "extraordinary reaction" to the explosive case: herein infinite PhDs lie.

Returning to the specific situation from the link, note that the price in eq. $(3.7)$ is determined by the ratio of two such infinite sums. Even from a mathematical point of view, this is a bit more favorable, because even if each sum separately diverges, their ratio may converge. But again, my point is, "don't let the mathematics make you forget the economics" of the situation.