Error in linear regression Given two series $(x_1,...x_n)$ and $(y_1,...y_n)$, and assume that we know $x_{n+1}$.
Given the fact that the pearson correlation won't change in the next observation of $y_{n+1}$, can we bound the error of predicting $y_{n+1}$ in terms of pearson correlation of two series?    
 A: As said in comments, your question is not entirely clear. You say "Given the fact that the pearson correlation won't change in the next observation of $y_{n+1}$", but correlation do not apply to a single observation, it applies to some well-defined population (or model). But maybe you try to say that the model do not change, that still do not give information about which model you have in mind. 
If what you have in mind is a standard linear model (which would be my guess), then you can find an answer here:  Obtaining a formula for prediction limits in a linear model  which tells you how to calculate prediction intervals in linear regression. The same principles (but not formulas) can be used for other kinds of models. For the difference between confidence intervals and prediction intervals see Difference between confidence intervals and prediction intervals or Linear regression prediction interval
A: Correlation is not sufficient to describe the distribution of the datapoints. You would need the full distribution in order to make predictions about the distribution of $y_{n+1}$ given $x_{n+1}$.
Anscombe's quartet is a good example of the different types of relationships that you can have while the correlation is the same. In all the four images below the correlation is the same 0.816

