Question 1177400
Let's solve this problem step-by-step.

**1. Define the Expected Value**

The expected value of a discrete random variable X is defined as:

* E(X) = Σ [x * P(X = x)]

In our case, X takes values 2^n, and P(X = 2^n) = 1/2^n. So:

* E(X) = Σ [2^n * (1/2^n)]  for n = 1, 2, 3, ...

**2. Simplify the Expression**

* E(X) = Σ [2^n / 2^n]
* E(X) = Σ [1] for n = 1, 2, 3, ...

**3. Analyze the Sum**

The sum is:

* E(X) = 1 + 1 + 1 + 1 + ...

This is an infinite sum of 1's.

**4. Conclusion**

Since we are adding 1 infinitely many times, the sum diverges to infinity.

* E(X) = ∞

Therefore, the expected value of X is infinite, meaning X has no mathematical expectation.