Let's break down the transformation from the first expression to the second step-by-step.

First Image Explanation

The expression in the first image is: $\sum_{n=0}^{\infty} \frac{e^{j\omega_0 n} + e^{-j\omega_0 n}}{2} \cdot e^{j\omega n}.$

1. Trigonometric Identity: The sum inside the fraction can be recognized as a trigonometric identity for the cosine function: $\frac{e^{j\omega_0 n} + e^{-j\omega_0 n}}{2} = \cos(\omega_0 n).$ However, in this case, instead of simplifying directly to a cosine, the next step involves manipulating the exponential terms.

2. Combining Exponentials: The product $\frac{e^{j\omega_0 n} + e^{-j\omega_0 n}}{2} \cdot e^{j\omega n}$ can be rewritten by distributing the exponential term: $\frac{e^{j(\omega_0 + \omega) n} + e^{-j(\omega_0 - \omega) n}}{2}.$

Second Image Explanation

The next step shown in the second image simplifies the summation using the new exponential terms: $\frac{1}{2} \left[ \sum_{n=0}^{\infty} e^{-j(\omega_0 + \omega) n} + \sum_{n=0}^{\infty} e^{j(\omega_0 - \omega) n} \right].$

1. Breaking Down the Exponential Sums: The expression has been broken into two separate geometric series:

   • The first series: $\sum_{n=0}^{\infty} e^{-j(\omega_0 + \omega) n}$.
   • The second series: $\sum_{n=0}^{\infty} e^{j(\omega_0 - \omega) n}$.

2. Geometric Series: Both of these summations are geometric series of the form $\sum_{n=0}^{\infty} r^n$, where $r = e^{-j(\omega_0 + \omega)}$ for the first sum and $r = e^{j(\omega_0 - \omega)}$ for the second sum.

Summary of the Transformation

The key transformation involves:

1. Distributing $e^{j\omega n}$ into the sum.
2. Recognizing that the exponential terms can be combined and rewritten.
3. Splitting the resulting sum into two geometric series for simplicity.

Would you like further details or have any other questions? Here are some follow-up questions to expand on this concept:

1. How do you simplify the geometric series in the final step?
2. What is the convergence condition for these geometric series?
3. How would this process change if $\omega_0$ and $\omega$ were real numbers rather than complex exponentials?
4. Can this technique be generalized for other forms of exponential products?
5. How can this summation be interpreted in terms of signal processing?

Tip: When dealing with sums of exponentials, it's often useful to recall trigonometric identities and properties of geometric series to simplify the expressions.