Problem 8 - Calculus Early Transcendentals

Question

Consider the infinite series \(\sum_{k=1}^{\infty} \frac{1}{k} .\) Evaluate the first four terms of the sequence of partial sums.

Step-by-Step Solution

Answer

Answer: The first four terms of the sequence of partial sums are \(1, 1.5, 1.833,\) and \(2.083\).

Step 1: Understanding the sequence of partial sums

A sequence of partial sums is a sequence where each term is the sum of the first n terms of a series. For our case, we need to find the first four partial sums for the series \(\sum_{k=1}^{\infty} \frac{1}{k}\). So, we will find the sum of the first, second, third, and fourth terms of the series respectively.

Step 2: Finding the first partial sum (S1)

The first partial sum is simply the first term of the series when k=1. So, plug in k=1 into the series formula: \(S_1 = \frac{1}{1} = 1\)

Step 3: Finding the second partial sum (S2)

For the second partial sum, we will sum the first two terms of the series: \(S_2 = \frac{1}{1} + \frac{1}{2} = 1 + 0.5 = 1.5\)

Step 4: Finding the third partial sum (S3)

Now, for the third partial sum, we sum the first three terms of the series: \(S_3 = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} = 1 + 0.5 + \frac{1}{3} \approx 1.833\)

Step 5: Finding the fourth partial sum (S4)

Finally, for the fourth partial sum, we sum the first four terms of the series: \(S_4 = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = 1 + 0.5 + \frac{1}{3} + 0.25 \approx 2.083\) So, the first four terms of the sequence of partial sums for the given infinite series are \(1, 1.5, 1.833,\) and \(2.083\).