Question

Choose your test Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{1}{k^{1+p}}, p>0$$

Short answer

Question: Determine the convergence of the following infinite series: $$\sum_{k=1}^{\infty} \frac{1}{k^{1+p}}, p>0$$ Answer: The given infinite series converges.

Step-by-step solution

Apply the Integral Test

To apply the Integral Test, we will consider the following improper integral: $$\int_1^\infty \frac{1}{x^{1+p}}dx$$ If this integral converges, so does the series, and if it diverges, the series also diverges.

Evaluate the integral

To solve the improper integral, evaluate the antiderivative and find the limit: $$\int_1^\infty \frac{1}{x^{1+p}}dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^{1+p}}dx $$ We can evaluate the integral using power rule for integration: $$-\frac{1}{p}x^{-p}\Big|_{1}^{b} = \lim_{b \to \infty} \left[ -\frac{1}{p}b^{-p} + \frac{1}{p}\right]$$

Analyze the limit

To determine the convergence of the series, analyze the limit: \begin{equation} \lim_{b \to \infty} \left[ -\frac{1}{p}b^{-p} + \frac{1}{p}\right] \end{equation} Since \(p>0\), the term \(-\frac{1}{p}b^{-p}\) goes to 0 as b approaches infinity. Thus, the limit is: $$\lim_{b \to \infty} \left[ -\frac{1}{p}b^{-p} + \frac{1}{p}\right] = 0 + \frac{1}{p}$$ Since, the limit is a finite value, the improper integral converges.

Conclude about the series' convergence

As per the Integral Test, since the improper integral \(\int_1^\infty \frac{1}{x^{1+p}}dx\) converges, the given series: $$\sum_{k=1}^{\infty} \frac{1}{k^{1+p}}, p>0$$ also converges.

Key concepts

Convergence of Series

Understanding the convergence of series is crucial in mathematical analysis. A series is a sum of an infinite sequence of terms. Not all series converge, which means they do not always come to a finite sum. To determine whether a series converges, we can apply various tests, such as the Integral Test. If we find that the sum of the series approaches a specific number, then the series is said to converge. If it does not, the series diverges. In the exercise, we applied the Integral Test to determine the convergence of the series \( \sum_{k=1}^{\infty} \frac{1}{k^{1+p}} \), where \(p>0\). The outcome was that the series converges because the corresponding improper integral converged to a finite value.

Improper Integral

An improper integral is an integral with at least one of its limits extending towards infinity or having an integrand that approaches infinity within the interval of integration. In the discussed exercise, the improper integral was \( \int_1^\infty \frac{1}{x^{1+p}}dx \). To determine the convergence of this integral, we found its antiderivative and then evaluated the limit as the upper limit of integration approaches infinity. An improper integral converges when the limit of the integral as the upper boundary approaches infinity results in a finite value. In this case, because the limit was finite, we concluded that the improper integral and hence, the series, converges.

Power Rule for Integration

The power rule for integration is a key tool in calculus that simplifies the process of finding antiderivatives. It states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), provided \(n eq -1\). In the context of our problem, we applied the power rule to find the antiderivative of \(\frac{1}{x^{1+p}}\). Although the function was presented in a fraction form, it could be rewritten as \(x^{-(1+p)}\). By applying the power rule, we derived the expression \(-\frac{1}{p}x^{-p}\), demonstrating how the power rule is a straightforward technique to solve integrals, and thus plays a crucial role in analyzing the convergence of our series.

Infinite Series

Infinite series are sums that contain an infinite number of terms. These series can either converge to a specific finite value or diverge, spreading out towards infinity or alternating indefinitely. The analysis of infinite series is a significant aspect of calculus and mathematical analysis because it helps in understanding the behavior of functions and sequences over an extensive range. In our given exercise, the series in question was \( \sum_{k=1}^{\infty} \frac{1}{k^{1+p}} \). It is an example of an infinite series because it has infinite terms starting from \(k=1\) up to infinity. Through application of tests like the Integral Test, mathematicians can determine whether such series converge or not, which has practical applications in various fields such as physics and engineering.