91Ó°ÊÓ

In calculus, an improper integral extends the concept of an integral to functions that have infinite intervals or integrands with infinite discontinuities. This can happen when the limits of integration are infinite, or if the function has vertical asymptotes within the given limits. To evaluate such integrals, limit processes are often used.

When dealing with an improper integral, you typically start by identifying if the bounds or the integrand cause the integrals' 'improperness'. In the example given with \[\int_{1}^{\infty} \frac{k}{\sqrt{k^2+4}} dk,\]we notice the upper limit is infinite, making it an improper integral. To evaluate this, we replace infinity with a variable, say \( t \), and evaluate the integral from 1 to \( t \). This is followed by taking the limit as \( t \) approaches infinity.

• Set up the integral with finite bounds.
• Evaluate the integral.
• Take the limit as the variable approaches infinity.

If the limit converges to a finite number, the integral (and series) converges. Otherwise, it diverges.

Series convergence is crucial in understanding whether a series sums up to a finite number or not. Consider the series \(\sum_{k=1}^{\infty} \frac{k}{\sqrt{k^{2}+4}}\) in the Original Exercise. The Integral Test provides a comparison between an infinite series and an improper integral.

To apply the Integral Test:

• Find a continuous, positive, and decreasing function \( f(x) \) analogous to the terms of the series.
• Compute the corresponding improper integral, along with its convergence.
• Conclude the behavior of the series based on the integral's outcome.

In the problem, since the related improper integral was shown to diverge, it is concluded that the series \(\sum_{k=1}^{\infty} \frac{k}{\sqrt{k^{2}+4}}\) also diverges.

Function continuity is one of the conditions needed for the Integral Test to apply when determining the convergence of a series. It assures that the function is smooth and without breaks over the interval of interest. A function \( f(x) \) is continuous if there are no gaps, jumps, or points where it is undefined.

Taking the function \( f(k) = \frac{k}{\sqrt{k^2+4}} \), we observe that:

• It is a rational function, combining polynomials and a square root.
• There are no points where the denominator becomes zero, meaning no singularities or undefined points.
• Since it remains defined for all \( k > 0 \), it is considered continuous over this interval.

Ensuring function continuity is necessary to plausibly link the series behavior to the corresponding improper integral.