Why study difference equation, sequences, etc? A couple of weeks ago I studied a thorough(yet basic) course in ODE, part of an course in analysis. Shortly after, we moved on to the study of difference equations, which is very much similar to differential equation. I am however unable to see the benefit of this theory since the solutions to such equations gives a function $ f:\mathbb{N}\rightarrow \mathbb{R} $ (the variable is discrete) and hence gives us less information than the equivalent solution to the differential equation.
The same thing can be said about series(compare to the integral). Can someone explain this?
 A: I was going to write this a comment, however it became too big to fit in the comment box:
How about this: the solutions of differential equations are usually required to be continuous (in particular, they are required to absolutely continuous in order to make sense of the differential equation). So if you'd like to model how something in real life evolves that, by definition, is an integer quantity (the number of molecules in a beaker, the size of a population, etc.), then ODEs might not be the best tool to do so (since their solutions cannot just jump from one integer value to another). 
Furthermore, think about the modelling process itself. Assume you're working from first principles. Then to construct such a model you will have to use the appropriate physical laws come up with a function $f:\mathbb{R}\to\mathbb{R}$ that takes in the quantity in question, $x$, and returns its derivative. But if $x$ must be an integer to make sense physically (say number of molecules in a beaker) how do you come up with $f(1/2)$?
Also, what if you'd like to run something on a computer? For example, virtually no ODEs have a closed form solution. So very often people find their solutions by "simulating" or "solving numerically" the ODEs on a computer. What this really means is that they construct difference equations whose solutions approximate very well those of the ODEs and then solve these difference equations on the computer.
On the technical side, sequences and series are a vital part of analysis. If nothing else, you need them to construct even basic calculus (the Riemann integral is defined using series). 
Anyway, I'm not sure what you're looking for here, so to sum up:


*

*Different situations call for different tools (whether to use ODEs or difference equations when constructing a model much depends on what the model is supposed to represent).

*Series and sequences and other "discrete-type" concepts are important to construct other more complicated concepts in analysis.

A: This is mainly a remark about your "contains more information" explanation in the comment.
If I understand correctly, you are trying to compare solutions for ODE, which you denote as $f\colon\bf R\to R$ and solutions to difference equations, which let me denote as $g\colon \bf N\to R$. Given that $\bf N$ is a subset of $\bf R$ you jump to the conclusion that the former have more information. This is not the case. In particular, a simple exercise is to understand and for any differential equation on $n$-dimensional state space it is possible to construct a discrete dynamical system, which would coincide with the flow of the ODE at discrete points. Opposite is not true. Think about a simple logistic map:
$$
g_\lambda\colon x\mapsto \lambda x(1-x).
$$
If you saw it before, probably you know that this discrete map can produce oscillations and chaotic trajectories (if you did not see it, put, e.g., $\lambda=3.789$ and iterate). No ODE of the first order can produce anything like this. Even autonomous  ODE of the second order will not produce chaos. Because there are quite strict requirements on the geometry of the phase orbits. 
