Math Problem Statement

The problem discusses the Integral Test for series convergence and the related criteria.

Solution

The image you provided contains a mathematical problem related to convergence criteria, particularly in the context of series.

Problem Overview:

The problem discusses the Integral Test for series convergence. Specifically, it states that if f(x)f(x) is a continuous, positive, decreasing function on the interval [1,)[1, \infty) and an=f(n)a_n = f(n), then the series n=1an\sum_{n=1}^{\infty} a_n converges if and only if the improper integral 1f(x)dx\int_{1}^{\infty} f(x) \, dx converges.

Key Points:

  1. Integral Test:

    • The convergence of a series n=1an\sum_{n=1}^{\infty} a_n is linked to the convergence of the integral 1f(x)dx\int_{1}^{\infty} f(x) \, dx.
    • The function f(x)f(x) must satisfy the conditions of being continuous, positive, and decreasing for the test to be applicable.
  2. Application of the Test:

    • You must verify whether f(x)f(x) meets the criteria.
    • Compute the improper integral 1f(x)dx\int_{1}^{\infty} f(x) \, dx and assess its convergence.
  3. Examples:

    • The test is likely applied to different functions f(x)f(x), showing whether the corresponding series converges or diverges.

Would you like to go through an example related to this, or need further details on any part of the problem?

Related Questions:

  1. What are the conditions necessary for the Integral Test to be valid?
  2. How can you determine if a function f(x)f(x) is decreasing on a given interval?
  3. What are the common functions used in conjunction with the Integral Test to assess series convergence?
  4. How does the result of the improper integral 1f(x)dx\int_{1}^{\infty} f(x) \, dx determine the behavior of the series?
  5. Can you apply the Integral Test to a series that does not start at n=1n=1?

Tip: Always check if the function f(x)f(x) is decreasing and

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Math Problem Analysis

Mathematical Concepts

Series Convergence
Integral Test
Improper Integrals

Formulas

Integral Test: \sum_{n=1}^{\infty} a_n converges if and only if \int_{1}^{\infty} f(x) dx converges

Theorems

Integral Test

Suitable Grade Level

Undergraduate Mathematics