91Ó°ÊÓ

When we study series, an important question arises: does the sum of its terms converge to a finite value, or does it diverge? In the context of mathematical analysis, convergence of a series means that as you add more and more terms, the total sum approaches a specific, finite number. This idea is crucial because it can tell us whether a series represents a stable sum or continues to grow indefinitely.
To determine convergence, there are several tests mathematical students often use, and one of the most approachable is the Integral Test. This test is particularly useful for series whose terms come from a function that is positive, continuous, and decreasing. In the exercise, we used the Integral Test to analyze the series \( \sum_{k=1}^{\infty} \frac{1}{\sqrt[3]{k+10}} \) by finding the corresponding integral and checking its behavior as it approaches infinity. Since the integral diverges, the series also diverges. This means the sum of the series becomes infinitely large.

Improper integrals often arise in calculus when you deal with integrals that have infinite limits or integrands (the function being integrated) that become infinite within the interval of integration. These integrals help us understand the behavior of functions over unbounded intervals or near points where the function increases without bound.
In the given exercise, we evaluated the improper integral \( \int_1^{\infty} \frac{1}{\sqrt[3]{x+10}} \, dx \). This is an example of an integral where the upper limit is infinity. By changing variables and simplifying, we found that as the new variable \( u \) tends to infinity, the integral does not approach a finite number; instead, it diverges. This behavior indicates the nature of the corresponding series because, according to the Integral Test, the convergence or divergence of the integral guides us to the same conclusion about the series.

A function is decreasing if, as you move to the right along the x-axis, the function values get smaller. This characteristic is significant in the application of the Integral Test. For the test to apply, the series' terms must form a function that is positive, continuous, and notably, decreasing.
In the context of our problem, the function \( f(x) = \frac{1}{\sqrt[3]{x+10}} \) was shown to be decreasing. This is because its derivative \( f'(x) \) is negative for all \( x \geq 1 \). Calculating the derivative, we found \( f'(x) = \frac{-1}{3(x+10)^{\frac{4}{3}}} \) and noted it was negative because the numerator is negative, and the denominator is always positive for \( x \geq 1 \). Thus, \( f(x) \) continually decreases, fulfilling one of the essential conditions for applying the Integral Test to decide on the divergence of the series.