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Chapter 8: Problem 8

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

Short Answer

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

Step by step solution

01

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.
02

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: \(S_1 = \frac{1}{1} = 1\)
03

Finding the second partial sum (S2)

For the second partial sum, we will sum the first two terms of the series: \(S_2 = \frac{1}{1} + \frac{1}{2} = 1 + 0.5 = 1.5\)
04

Finding the third partial sum (S3)

Now, for the third partial sum, we sum the first three terms of the series: \(S_3 = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} = 1 + 0.5 + \frac{1}{3} \approx 1.833\)
05

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 \(2.083\).

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