Consider the infinite series \( \sum_{\mathrm{k}=1}^{\infty} \frac{1}{\mathrm{k}} \). Evaluate the first four terms of the sequence of partial sums. The first partial sum is \( \mathrm{S}_{1}=\square \). (Type an integer or a simplified fraction.) The second partial sum is \( \mathrm{S}_{2}=\square \). (Type an integer or a simplified fraction.) The third partial sum is \( \mathrm{S}_{3}=\square \). (Type an integer or a simplified fraction.) The fourth partial sum is \( \mathrm{S}_{4}=\square \). (Type an integer or a simplified fraction.)

To evaluate the first four terms of the sequence of partial sums for the infinite series \( \sum_{k=1}^{\infty} \frac{1}{k} \), we need to compute the partial sums \( S_n = \sum_{k=1}^n \frac{1}{k} \) for \( n = 1, 2, 3, \) and \( 4 \). 1. The first partial sum is: \[ S_1 = \frac{1}{1} = 1 \] 2. The second partial sum is: \[ S_2 = \frac{1}{1} + \frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2} \] 3. The third partial sum is: \[ S_3 = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} = 1 + \frac{1}{2} + \frac{1}{3} = \frac{6}{6} + \frac{3}{6} + \frac{2}{6} = \frac{11}{6} \] 4. The fourth partial sum is: \[ S_4 = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \] To find a common denominator (12): \[ S_4 = \frac{12}{12} + \frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{25}{12} \] Thus, the values for the first four partial sums are: - \( S_1 = 1 \) - \( S_2 = \frac{3}{2} \) - \( S_3 = \frac{11}{6} \) - \( S_4 = \frac{25}{12} \) Final results: - The first partial sum is \( S_1 = 1 \). - The second partial sum is \( S_2 = \frac{3}{2} \). - The third partial sum is \( S_3 = \frac{11}{6} \). - The fourth partial sum is \( S_4 = \frac{25}{12} \).