Problem 17 - Thomas Calculus

Question

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty} \frac{n^{\sqrt{2}}}{2^{n}} $$

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Identify the type of series

The given series is a sum of terms of the form \( \frac{n^{\sqrt{2}}}{2^n} \). This is a form of a general term making the series a type of exponential or power series, depending on how we observe the exponent's structure. Here, we see a power of \( n \) in the numerator and an exponential \( 2^n \) in the denominator.

Step 2: Choose a suitable test

Considering the form of the series, use the Ratio Test, which is effective for series with factorials, powers, or exponential functions. The test involves computing \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), where \( a_n = \frac{n^{\sqrt{2}}}{2^n} \).

Step 3: Calculate the ratio of successive terms

Find \( a_{n+1} = \frac{(n+1)^{\sqrt{2}}}{2^{n+1}} \). Now calculate the ratio:\[\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(n+1)^{\sqrt{2}}}{2^{n+1}} \cdot \frac{2^n}{n^{\sqrt{2}}} \right| = \left| \frac{(n+1)^{\sqrt{2}}}{n^{\sqrt{2}} \cdot 2} \right|. \] Simplify this to:\[ \frac{(n+1)^{\sqrt{2}}}{n^{\sqrt{2}} \cdot 2} = \frac{\left( 1 + \frac{1}{n} \right)^{\sqrt{2}}}{2}. \]

Step 4: Evaluate the limit of the ratio

Calculate the limit as \( n \to \infty \):\[\lim_{n \to \infty} \left( \frac{\left( 1 + \frac{1}{n} \right)^{\sqrt{2}}}{2} \right).\]Using the approximation \( \left( 1 + \frac{1}{n} \right)^{\sqrt{2}} \approx 1 + \frac{\sqrt{2}}{n} \to 1 \) as \( n \to \infty \),\[\lim_{n \to \infty} \left( \frac{1 + \frac{\sqrt{2}}{n}}{2} \right) = \frac{1}{2}.\]

Step 5: Apply the Ratio Test conclusion

Since the limit equals \( \frac{1}{2} < 1 \), the Ratio Test tells us that the series \( \sum_{n=1}^{\infty} \frac{n^{\sqrt{2}}}{2^n} \) converges. According to the Ratio Test, if the limit is less than 1, the series converges absolutely.

Key Concepts

Ratio Test

The Ratio Test is a powerful tool for determining the convergence or divergence of an infinite series, especially where terms involve factorials, powers, or exponentials. It helps assess the behavior of a series' terms as they extend towards infinity. To apply this test effectively, you examine the limit of the ratio between consecutive terms in the series. Let's say the series is represented by terms labeled as \( a_n \). The Ratio Test involves calculating the limit:

• \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

This limit helps determine the fate of the series:

• If the limit is less than 1, the series converges absolutely, meaning it converges very securely without any doubt.
• If the limit is greater than 1, the series diverges.
• If the limit is exactly 1, the test is inconclusive; hence, a different method must be used to assess convergence.

In the given exercise, applying the Ratio Test led to calculating:

• \[ \lim_{n \to \infty} \left( \frac{(1 + \frac{1}{n})^{\sqrt{2}}}{2} \right) = \frac{1}{2} \]

Since this value is less than 1, we conclude that the series converges at an absolutely reliable level.

Power Series

A power series is a type of infinite series related to functions involving a variable raised to successive integer powers, coupled with coefficients. They have the general form:

• \[ \sum_{n=0}^{\infty} a_n x^n \]

Here, each \( a_n \) acts as a coefficient in front of \( x^n \). Power series can represent a wide range of functions depending on their coefficients and radii of convergence—informing how far from the center of the series the function remains valid.
In context with this exercise, the series \( \sum_{n=1}^{\infty} \frac{n^{\sqrt{2}}}{2^n} \) serves as a variant involving powers of \( n \) with the denominator being dominated by an exponential. While not in the standard form of a power series, it exhibits characteristics akin to one, where its convergence can also be examined efficiently through convergence tests such as the Ratio Test.
Understanding power series is crucial:

• They represent functions in simple polynomial-like terms.
• Calculations of derivatives and integrals become straightforward.
• Power series extend functions analytically and apply significantly in calculus and analysis.

The exercise exemplifies a scenario where understanding and identifying the form of series aids in selecting the right method for convergence assessment.

Exponential Functions

Exponential functions are functions of the form \( f(x) = a \cdot b^x \), characterized by variable exponents. These functions grow or decay at constant rates and are inherently involved in many natural processes like population growth, radioactive decay, and compound interest.
The series in the exercise,

• \( \sum_{n=1}^{\infty} \frac{n^{\sqrt{2}}}{2^n} \)

highlights the interplay between powers of \( n \) in the numerator and exponential terms \( 2^n \) in the denominator. The structure of the denominator implies exponential reduction with increasing \( n \), causing rapid diminishment of terms when compared to the increasing numerator power.
Key attributes of exponential functions include:

• Rapid growth or decay depending on base \( b \); if \( b > 1 \), the function grows, if \( 0 < b < 1 \), it decays.
• Exponential terms power convergence properties, particularly when \( b > 1 \) in series, as seen in this exercise.

In applying convergence tests, particularly the Ratio Test, exponential elements provide clear indicators of how quickly series terms vanish, supporting or refuting convergence depending on their comparative strength to polynomial terms. This makes exponential functions important pivot points in series like the one in the exercise, dictating their general behavior.