1. The problem asks us to find the Moment Generating Function (MGF) for a random variable defined as $X = \frac{1}{k}$, where $k = 1,2,3,\ldots$. 2. Typically, the MGF of a random variable $X$ is defined as $$M_X(t) = E[e^{tX}] = \sum_x e^{tx} P(X=x)$$ where the expectation is taken over the probability distribution of $X$. 3. In this problem, no probability distribution for $k$ is given. To find the MGF, we need the probabilities $P(X=1/k)$ for each $k$. 4. Without a probability mass function (pmf) or some description of how $k$ is distributed, the MGF cannot be explicitly computed. 5. If we assume $k$ takes values $1,2,3,\ldots$ with probabilities $p_k$, then $$M_X(t) = \sum_{k=1}^\infty e^{t(1/k)} p_k$$ 6. The exact form depends on the choice of $p_k$. For example, if $k$ is geometric or any other discrete distribution, they must be specified. 7. Therefore, without further information on the distribution of $k$, the MGF cannot be determined. Final answer: The MGF for $X=1/k$ depends on the distribution of $k$. Please provide the probabilities $P(k = k)$ to proceed.