Problem 112 - College Algebra Essentials

Question

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} ; \ldots .\) Describe what you observe.

Step-by-Step Solution

Answer

The pattern observed is that the series is approximating towards the value of \(\ln 2 = 0.693\).

Step 1: Calculate ln 2 using a calculator

Using a scientific calculator, the value of \(\ln 2\) approximates to 0.693.

Step 2: Calculate the given expressions

First, calculate each of the expressions: \(1-\frac{1}{2} = 0.5\) \(1-\frac{1}{2}+\frac{1}{3} = 0.833\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} = 0.583\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} = 0.783\)

Step 3: Observe the pattern

It can be observed from the calculated values that the sequence is approaching the value of \(\ln 2\). As one further calculates more terms from the series, the results get even closer to \(\ln 2 = 0.693\). This illustrates the concept of a mathematical series, where the sum of the series tends towards a particular value.