Problem 55 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. Suppose that \(f\) is a continuous, positive, and decreasing function on $[1, \infty) .\( If \)f(n)=a_{n}\( for \)n \geq 1\( and \)\sum_{n=1}^{\infty} a_{n}$ is convergent, then $\sum_{n=1}^{\infty} a_{n} \leq a_{1}+\int_{1}^{\infty} f(x) d x$.

Step-by-Step Solution

Answer

The statement is true. We established that for each \(n\), the height of the rectangle is given by \(f(x)\geq a_{n+1}\), and thus the sum of the series \(\sum_{n=1}^{\infty} a_n\) is less than or equal to the sum of the area under the curve of the function \(f(x)\), represented as \(a_1 + \int_{1}^{\infty} f(x) dx\).

Step 1: First, let's think about the area under the curve \(f(x)\) that can be covered by rectangles of length 1. Each rectangle can be represented as \(a_n\) with \(n \geq 1\). Since \(f\) is positive and decreasing, the rectangles get smaller as we continue along the x-axis. Notice that for any value \(x \in [n,n+1)\) the height of the rectangle is given by \(f(x) \geq f(n+1) = a_{n+1}\). #Step 2: Compare the series and areas of the rectangles#

Let's compare the sum of the series and the areas of the rectangles: $$\sum_{n=1}^{\infty} a_n \leq a_1 + \sum_{n=1}^{\infty} \int_{n}^{n+1} f(x) dx$$ #Step 3: Write the sum using the integral#

Step 2: From the inequality we derived in Step 2, we can now write the sum as an integral: $$\sum_{n=1}^{\infty} a_n \leq a_1 + \int_{1}^{\infty} f(x) dx$$ #Conclusion#

Since we have shown that \(\sum_{n=1}^{\infty} a_{n} \leq a_{1}+\int_{1}^{\infty} f(x) dx\), the statement is true. The sum of the series is less than or equal to the sum of the area under the curve of the function \(f(x)\).