Using the Integral Test on Exponential Series

Try using the integral test on your own for the series $\sum_{n=1}^{\infty} e^n$ and determine if it converges or diverges.

The integral test is a powerful and often used method for determining the convergence or divergence of an infinite series. In this particular problem, you are asked to apply the integral test to the series whose general term is an exponential function of the form $e^n$. This exercise encourages you to sharpen your skills in recognizing when the integral test is appropriate and acting on these insights.

The concept behind the integral test is to compare the sum of an infinite series to a related improper integral. If the integral of the function corresponding to the series is finite and converges, then the series also converges. Conversely, if the integral is infinite and diverges, the series does as well. Therefore, in the case of the given series with base $e$, you should aim to set up the corresponding improper integral and evaluate its convergence properties.

This problem also highlights the broader concept of series convergence tests. Knowing a variety of tests allows you to choose the most efficient one based on the form of the series' general term. Through practicing with exponential series, you further develop your intuition in handling different series types, a critical skill in calculus and higher-level mathematics.

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