Convergence of an Infinite Series Involving 2n

Determine if the infinite series of $2n$ will converge or diverge.

When faced with the task of determining whether an infinite series converges or diverges, it's crucial to have a strategy in place for evaluating the behavior of its terms as n approaches infinity. Infinite series can vary greatly in form, making some easier to analyze than others. In this problem, we are considering a series involving terms of the form 2n, which is a linear sequence. For such problems, recognizing that the series is arithmetic is vital, as arithmetic sequences generally display linear growth, and their series do not converge.

To start, it's important to apply fundamental tests for convergence, such as the nth-term test, which can quickly help deduce if the series diverges. If the terms of the sequence do not approach zero as n increases, the series cannot converge. For arithmetic sequences, each term increases by a constant amount, leading the series to diverge since the terms grow indefinitely rather than shrink towards zero. Therefore, using direct reasoning about the nature of the sequence—or by employing the nth-term divergence test—one can ascertain divergence swiftly.

Understanding divergence through this kind of analysis is an integral skill in calculus, especially within the broader study of series. While divergent series are often less useful for practical applications involving sums, recognizing their divergent nature helps prevent the misapplication of convergent series techniques, which could lead to incorrect conclusions. This problem, while straightforward, reinforces the critical thinking skills needed to analyze patterns and behaviors of mathematical sequences and series.

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