Convergence of the Series Sum 1n2 Using Integral Test

Explore whether the infinite series from n equals 1 to infinity of $\frac{1}{n^2}$ converges or diverges using the integral test.

The problem asks us to examine the convergence of the infinite series given by the sum of 1 over n squared by using the integral test. This series is a specific example of the p-series, which is known to converge if the power p is greater than 1. The integral test is a powerful tool in the study of series, as it allows us to determine convergence through the lens of improper integrals. By evaluating the improper integral that corresponds to the general term of our series, we can draw definitive conclusions about the series' behavior.

The integral test requires comparing our given series to an improper integral. To apply the test effectively, it's crucial to ensure that the function we're integrating is positive, continuous, and decreasing. In this case, the function 1 over x squared meets these criteria for x greater than or equal to 1. By integrating this function from 1 to infinity, we observe that the integral converges to a finite value. This leads directly to the conclusion that our original series also converges.

Understanding the integral test not only helps with series analysis but also deepens comprehension of the relationship between discrete sums and continuous integrals. This problem, therefore, offers insights that are integral to mastering concepts in calculus, particularly in the study of convergence and divergence of series. In practice, being able to interchangeably use series and integrals in mathematical and real-world applications is a testament to the power and utility of calculus. Through such exercises, one gains a greater appreciation for the elegant ways in which calculus allows us to explore and understand mathematical infinity.

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