Problem 27 - Essential Calculus

Question

Find the limit (if possible) of the sequence. \(a_{n}=\frac{2 n}{\sqrt{n^{2}+1}}\)

Step-by-Step Solution

Answer

The limit of the sequence as n approaches infinity is 2.

Step 1: Identify the highest power of n in the denominator

In this case, the highest power of n in the denominator is \(n^{2}\). We shall divide all terms by \(n^{2}\) to simplify the function.

Step 2: Divide all terms by \(n^{2}\)

The sequence becomes \( \frac{2}{\sqrt{1+1/n^{2}}}\) when all terms are divided by \(n^{2}\).

Step 3: Apply the rules of limits to the sequence

As n approaches infinity, \(1/n^{2}\) approaches 0 while the constant remains unaffected. So, the sequence simplifies to \( \frac{2}{\sqrt{1+0}}\), which simplifies further to 2.

Key Concepts

Limit of a Sequence

In calculus, the "limit of a sequence" is the value that the terms of a sequence approach as the index (usually represented by \( n \)) goes to infinity. Understanding the limit of a sequence is crucial because it allows us to predict the behavior of sequences as they progress indefinitely. To find the limit, we typically manipulate the sequence into a simpler form where the behavior at infinity becomes clear. For example, in the given sequence \( a_{n} = \frac{2n}{\sqrt{n^{2} + 1}} \), we simplified the expression by dividing both the numerator and the denominator by \( n \). This strategy helps us focus on the terms that grow or shrink significantly as \( n \) increases.Often, sequences approach a fixed number, like in this example, which simplifies to 2 as \( n \) becomes very large.

Convergence

Convergence in sequences refers to a sequence's behavior as its terms get infinitely close to a specific value called the "limit." For a sequence to be convergent, its terms must approach this limit as the sequence progresses. The limit must exist and must be a finite number.In simpler terms, if you keep picking larger and larger values of \( n \), the terms become almost indistinguishable from the limit.In the example given, \( a_{n} = \frac{2n}{\sqrt{n^{2} + 1}} \) converges to 2. As \( n \) increases, the terms in the sequence stabilize around the number 2. For students, ensuring sequences converge often involves isolating the dominant terms, as shown, and confirming they approach a finite value.

Infinity in Sequences

"Infinity in sequences" is a concept that indicates the absence of bounds as the index \( n \) of a sequence increases. When evaluating the sequence approach towards infinity, it's essential to determine how the sequence behaves in the long run.In the context of limits and convergence, going to infinity helps us define the terms' behavior beyond all integer boundaries. To manage infinity effectively, we often express sequences in simpler forms, highlighting terms that diminish or grow as \( n \) becomes very large. In our sequence \( a_{n} = \frac{2n}{\sqrt{n^{2} + 1}} \), examining the behavior as \( n \) approaches infinity gives insight into its eventual approach to a limit, which is 2.Understanding infinity in this sense aids in predicting patterns and verifying that certain real numbers are approached consistently.