   Chapter 11.3, Problem 2E

Chapter
Section
Textbook Problem

# Suppose f is a continuous positive decreasing function for x ≥ 1 and an = f(n). By drawing a picture, rank the following three quantities in increasing order: ∫ 1 6 f ( x ) d x     ∑ i = 1 5 a i     ∑ i = 2 6 a i

To determine

The rank of the three quantities are 16f(x)dx,i=15ai and i=26ai .

Explanation

Result used: Remainder Estimate for the Integral Test

If the function f(x) is continuous, positive, and decreasing for xn and an is convergent and Rn=ssn , then n+1f(x)dxRnnf(x)dx .

Given:

The function f(x) is continuous, positive, and decreasing for n1 and an=f(n) .

Calculation:

From the graph below, the total area of the rectangles is more than the area under f(x) of  the curve.

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