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:

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