Problem 67 - Essential Calculus

Question

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}(2 \sqrt[n]{n}+1)^{n} $$

Step-by-Step Solution

Answer

The series \(\sum_{n=1}^{\infty}(2 \sqrt[n]{n}+1)^{n}\) is divergent.

Step 1: Apply the Root Test

The expression of which we need to find the nth root is \( |(2 \sqrt[n]{n}+1)^{n}| \). This simplifies to \( |2 \sqrt[n]{n} + 1| \) because the nth root and the nth power cancel each other out.

Step 2: Calculate the limit

Find \(\lim_{n \to \infty} (|2 \sqrt[n]{n} + 1|)\). As n approaches infinity, \( \sqrt[n]{n} \) approaches 1. So, \(\lim_{n \to \infty} (|2 \sqrt[n]{n} + 1|) = |2*1 + 1| = 3\)

Step 3: Determine convergence or divergence

By the Root Test, the series is convergent if the limit found is less than 1, divergent if it's greater than 1, and the test is inconclusive if it equals 1. Here, since the limit is 3 which is greater than 1, we can conclude that the series \(\sum_{n=1}^{\infty}(2 \sqrt[n]{n}+1)^{n}\) is divergent.

Key Concepts

Convergence and Divergence of Series

In mathematics, the concepts of convergence and divergence help us understand whether a series sums to a finite value or not.
A series is simply the sum of a sequence of numbers. If this sum approaches a certain finite value as more terms are added, we say the series is convergent.

• Convergent Series: The terms of the series approach zero, and the total sum of the series approaches a constant value.
• Divergent Series: The terms do not approach zero or the total sum increases indefinitely, meaning no finite sum can be determined.

For example, in the given series \( \sum_{n=1}^{\infty}(2 \sqrt[n]{n}+1)^{n} \), we used the Root Test to find out whether it is convergent or divergent by calculating the limit of its terms.By examining the behavior of the terms of the series, we determined that it was divergent because the limit was greater than 1. Understanding convergence helps ensure calculations related to infinite series are accurate when applied to real-world problems like physics and finance.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus, crucial for determining the behavior of series and functions as a variable approaches some value.
A sequence is a list of numbers in a specific order, usually defined by a rule or formula. The limit of a sequence is the number the terms of the sequence get closer to as the sequence progresses to infinity.

• If the terms approach a single number, the sequence has a limit.
• If they do not approach a single number, or keep growing, the sequence has no limit.

In our exercise, we computed \( \lim_{n \to \infty} (|2 \sqrt[n]{n} + 1|) \).As \( n \) approaches infinity, \( \sqrt[n]{n} \) approaches 1, making the entire expression simplify to \( |2 \/times 1 + 1| = 3 \). This limit tells us about the behavior of the sequence, and ultimately the series it defines.

Nth Root

The concept of the nth root is important in both algebra and calculus, particularly for analyzing sequences and series.
The nth root of a number is the value that, when raised to the nth power, gives the original number.

• If \( a^n = b \), then \( a \) is the nth root of \( b \).
• For example, the square root is the 2nd root, the cube root is the 3rd root, etc.

In our original problem, the expression contained \( 2 \sqrt[n]{n} \).Evaluating this as \( n \) approaches infinity:

• The expression \( \sqrt[n]{n} \) simplifies to 1 because the values become closer to 1 with increasing \( n \).

Thus, recognizing how the nth root behaves as \( n \) grows is critical for understanding the tendencies of sequences. Such insights help in utilizing tests like the Root Test to assess the convergence or divergence of more complex series.