Convergence of the Sequence A sub N equals N over N plus One

Determine if the sequence $A_n = \frac{N}{N+1}$ is convergent or divergent as $N \to \infty$.

When examining the convergence or divergence of a sequence, one of the fundamental concepts involves analyzing the behavior of the sequence as its index approaches infinity. In this problem, the sequence in question is defined as $A_n = \frac{N}{N+1}$. The strategy to determine convergence is rooted in the comparison of the given sequence with a simpler or well-known sequence whose behavior at infinity is understood.

One useful approach is to consider the limiting behavior of the sequence as N becomes very large. Intuitively, you can simplify the expression by dividing each term by N, which can often highlight the dominant behavior between terms. This can simplify the sequence to a form easier to recognize and assess for limits, typically reducing it to a constant plus a diminishing fraction as N approaches infinity. In this particular problem, determining whether the terms of the sequence approach a finite limit can reveal the nature of convergence.

The underlying concepts also tie into the broader topic of sequences in calculus, where a thorough understanding of limits is fundamental. Investigating the long-term behavior of sequences is a stepping stone to understanding series later on, where the sum of infinite sequences comes into play. A solid grasp of convergence lays the groundwork for more advanced concepts like power series or Taylor series, which rely heavily on these principles.

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