You are trying to determine whether diverges. Which integral can you use to give a lower bound to the sequence of partial sums ?

A:

B:

C:

D:

Question

You are trying to determine whether  diverges. Which integral can you use to give a lower bound to the sequence of partial sums ?

A:

B:

C:

D:

BlackTom AI · Accepted Answer

The question asks about using an integral to obtain a lower bound for the sequence of partial sums of a series whose terms are presumably positive and decreasing, in order to assess divergence.

First, recall the integral test for a positive, decreasing function f with a_n = f(n): for each N, the partial sum S_N = ∑_{n=1}^N a_n satisfies a pair of inequalities that relate it to integrals of f:
- A standard form gives: ∑_{n=1}^N f(n) ≥ ∫_{1}^{N+1} f(x) dx.
- Conversely, ∑_{n=1}^N f(n) ≤ ∫_{0}^{N} f(x) dx when f is monotone decreasing on [0, ∞).
These inequalities imply that the integral ∫_{1}^{N+1} f(x) dx serves as a lower bound for the partial sums S_N.

Therefore, to obtain a lower bound on the partial sums and hence to study divergence, you should use the integral from 1 to N+1 of f(x) dx (i.e., ∫_{1}^{N+1} f(x) dx). If this integral diverges as N → ∞, then the sequence of partial sums S_N also diverges.

Since the answer options A–D are not provided here, I’m focusing on the standard result: the lower bound for S_N is given by ∫_{1}^{N+1} f(x) dx for a decreasing positive f, which is the typical integral used in the integral test to prove divergence.

If any of the options correspond to ∫_{1}^{N+1} f(x) dx as the lower bound, that would be the correct choice under the usual integral-test framework. If the options instead use a different interval (e.g., ∫_{1}^{N} or ∫_{0}^{N} for a lower bound), those would generally be either weaker bounds or applicable to a different inequality form and should not be chosen when the goal is a direct lower bound for S_N as N grows.

MAT187H1 F LEC0101 Integral Test for Series (Pre-Class Essentials)

You are trying to determine whether diverges. Which integral can you use to give a lower bound to the sequence of partial sums ? A: B: C: D:

View Explanation

Log in for full answers

Similar Questions

Which of the following series can you use the integral test to determine if they converge? Select all that apply. A: B: C: D:

Let 𝑓 be a CONTINUOUS, POSITIVE, DECREASING function on [ 0 , ∞ ) . Which of the following statements must be true? Select all the correct answers.

What is the result of applying the integral test to the series [math: ∑n=1∞12n−1] \displaystyle\sum\limits_{n=1}^ \infty \frac{1}{2n-1} ?

In a consumer society, many adults channel creativity into buying things

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler