DTFT vs the spectrum of a sampled signal

[CoolDude420]

Homework Statement

Hi,

So we started sampling/sampling theorem, dirac delta, DTFT in a digital signal processing module and I'm kinda confused. I understand how to derive the following formulae but these two formulae are so different to each other that I don't understand why?

We are first told that the spectrum of the sampled signal is,

I completely understand where this comes from (the whole dirac delta part/sifting property and FT property). Okay all makes sense now. Then we are introduced to the DTFT and again I understand the derivation but the result is really different,

Now, how can these two things be equal to each other? I mean in both cases, we were deriving the frequency spectrum of the sampled signal. Why are the answers so different?

Homework Equations

The Attempt at a Solution

 

[scottdave]

You understand what $e^{jθ}$ represents? It is $cosθ + j sinθ$. The f_n coefficients are such that the complex terms cancel out so that you have a real signal.

I don't mean to be rude, I'm just trying to figure what you know.

 

[CoolDude420]

You understand what $e^{jθ}$ represents? It is $cosθ + j sinθ$. The f_n coefficients are such that the complex terms cancel out so that you have a real signal.

I don't mean to be rude, I'm just trying to figure what you know.

So essentially these two are equivalent? Would I be right in saying that the first equation is sort of good for illustration purposes and the second one helps in leading to the DFT for actually calculating?