91Ó°ÊÓ

When we talk about convergence in the context of improper integrals, we are asking whether the integral settles to a finite value as one of its limits approaches infinity. If it does, we say the integral converges. If it doesn't, it diverges.
To determine convergence, we often use techniques involving limits. By expressing the improper integral as a limit, we can evaluate whether it approaches a specific value.
For example, in the given problem, we transformed the improper integral \(\int_{1}^{\infty} \frac{3 dx}{x^{2}+1}\) into \(\lim_{b \to \infty} \int_{1}^{b} \frac{3 dx}{x^{2}+1}\). This step helps us check the behavior of the integral as the upper limit grows without bound.
The limit ultimately calculated to be \(\frac{3\frac{\pi}{2}}{4}\), which is a finite number, indicating convergence. If the result were infinity or did not exist, the integral would be divergent.

Solving improper integrals requires knowledge of various integration techniques. These are methods used to find the antiderivative of a function, which is key to evaluating definite integrals.
In our example, the integrand \(\frac{3}{x^{2}+1}\) is particularly straightforward because we recognize that the derivative of the arctangent function \(\arctan(x)\) is \(\frac{1}{x^{2}+1}\). Therefore, we can write:
\(\int \frac{3 dx}{x^{2}+1} = 3 \arctan(x)\).
Once we have this antiderivative, the next step is to apply the limits from 1 to some upper bound b (which approaches infinity).
Effective integration techniques include:

• Recognizing standard integral forms (like \(\frac{1}{x^{2}+1}\) for \(\arctan(x)\))
• Substitution methods
• Partial fraction decomposition
• Integration by parts

For improper integrals specifically, transforming them into limits as shown is critical.

An infinite series sums an infinite number of terms. The series in the problem \(\sum_{n=1}^{\infty} \frac{3}{n^{2}+1}\) is closely related to the improper integral we just evaluated.
To determine if an infinite series converges, we can often compare it to a known convergent series. Here, we used the fact that:

• \(\frac{3}{n^{2}+1}\) resembles \(\frac{3}{n^{2}}\)
• The series \(\sum_{n=1}^{\infty} \frac{3}{n^{2}}\) converges because it is a p-series with p = 2, which is greater than 1.

By noting that \(\frac{3}{n^{2}+1}\) is always less than \(\frac{3}{n^{2}}\), we deduce the given series also must converge.
There are various tests to check for the convergence of series such as:

• Comparison Test (as used here)
• Ratio Test
• Root Test
• Integral Test
• Alternating Series Test

Understanding these tests is crucial for analyzing series effectively.