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 $f(x)$ is a continuous, positive, decreasing function on the interval $[1, \infty)$ and $a_n = f(n)$, then the series $\sum_{n=1}^{\infty} a_n$ converges if and only if the improper integral $\int_{1}^{\infty} f(x) \, dx$ converges.

Key Points:

1. Integral Test:

   • The convergence of a series $\sum_{n=1}^{\infty} a_n$ is linked to the convergence of the integral $\int_{1}^{\infty} f(x) \, dx$.
   • The function $f(x)$ 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 $f(x)$ meets the criteria.
   • Compute the improper integral $\int_{1}^{\infty} f(x) \, dx$ and assess its convergence.

3. Examples:

   • The test is likely applied to different functions $f(x)$, 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 $f(x)$ 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 $\int_{1}^{\infty} f(x) \, dx$ determine the behavior of the series?
5. Can you apply the Integral Test to a series that does not start at $n=1$?

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