Convergence of Series with Comparison Test

Using the comparison test, determine if the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^n + 1}$ is convergent by comparing it to the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^n}$.

In this problem, we explore the convergence of an infinite series using the comparison test, a fundamental concept in the study of series convergence. The comparison test allows us to determine the convergence or divergence of a given series by comparing it to another series whose behavior is already known. This technique is particularly useful when dealing with series that are difficult to analyze directly.

The given series involves a term with both an exponential function and a constant addition within the denominator. The key to applying the comparison test is to find a benchmark series with a similar form, typically one that simplifies the problem and has known convergence properties. In this case, the series to compare with is geometric, a standard benchmark for applying the comparison test. Understanding geometric series and their convergence properties, as well as being able to manipulate series terms to fit the comparison test framework, are crucial skills in series analysis.

By mastering the steps and logic of the comparison test, students enhance their problem-solving toolkit for tackling convergence issues. This involves not only picking suitable comparison series but also rigorously justifying why the particular choice of comparison leads to valid conclusions about the original series. This is a skill widely applicable in calculus and analysis, enabling the solving of complex problems encountered in advanced mathematics education.

Related Problems