Convergence of Series Using the Integral Test

Calculate whether the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ is convergent using the integral test, and estimate its sum.

The convergence of a series is a fundamental concept in calculus, as it determines whether the infinite sum of its terms approaches a finite value. In this problem, we explore the convergence of the series using what is known as the integral test. The integral test provides a valuable connection between sums and integrals, allowing us to use techniques from integration to determine the behavior of series. The integral test requires the function associated with the series terms to be positive, continuous, and decreasing. By comparing the series to an improper integral, we can make clear conclusions about its convergence or divergence.

In this specific problem, the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ is analyzed. This is a classic example often examined in calculus courses because of its well-known convergence. By applying the integral test, we calculate the integral of the function $f(x) = \frac{1}{x^2}$ from 1 to infinity. If this integral converges, then the corresponding series also converges. Additionally, estimating the sum of the series involves understanding how closely the sum of its initial terms approximates the value of the infinite sum. Through estimation techniques, students can grasp the behavior of series at a deeper level, enhancing their computational skills and analytical thinking.

Related Problems