Series Convergence with Integral Test

Determine whether the series represented by the function is convergent or divergent using the integral test for the function: $\displaystyle \int_{1}^{\infty} \frac{1}{2^x} \, dx$

In this problem, we are tasked with determining the convergence or divergence of a series using the integral test. The integral test is a powerful technique that helps relate infinite series to improper integrals. In this context, it provides a bridge connecting discrete sums (series) with continuous accumulation represented by integration. The series in question is analyzed by evaluating an improper integral, which requires careful assessment of the integral's behavior as its bounds approach infinity.

The integral test relies on evaluating the convergence of the associated improper integral and provides conclusive evidence about the series’ convergence if applicable. However, it is crucial to ensure the function meets the conditions necessary for the integral test: the function must be continuous, positive, and decreasing. The integral test can only be applied if these conditions are satisfied over the interval from one to infinity. In this case, we analyze the behavior of the function 1 divided by 2 to the power of x as x goes to infinity, discussing both its treatment as an exponential decay function and its integration limits.

Moreover, while solving this problem, recognizing the decay rate or terms might provide insights or shortcuts. Through the lens of the integral test, we closely examine how the improper integral behaves, understanding when it converges to a finite limit, and thereby when the respective series converges. These concepts not only aid in the understanding of series and sequences but also reveal connections between discrete summations and continuous integrals in calculus.

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