Problem 81 - Essential Calculus

Question

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=1}^{\infty} \frac{10 n+3}{n 2^{n}} $$

Step-by-Step Solution

Answer

According to ratio test, the series \( \sum_{n=1}^{\infty} \frac{10 n+3}{n 2^{n}} \) converges.

Step 1: Identify the Series Terms

Identify the terms of the series. In this case, the general term of the given series is \( a_n = \frac{10n+3}{n2^n} \). The next term \( a_{n+1} = \frac{10(n+1)+3}{(n+1)2^{n+1}} \)

Step 2: Apply Ratio Test

Apply the ratio test, which is calculating the limit of the absolute ratio of consecutive terms as \( n \) tends to infinity. \( \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|=\lim_{n \to \infty} \left|\frac{\frac{10(n+1)+3}{(n+1)2^{n+1}}}{\frac{10n+3}{n2^n}}\right| \)

Step 3: Simplify the Ratio

Simplify the ratio. \( \lim_{n \to \infty} \left|\frac{10(n+1)+3}{10n+3} \right| \times \frac{1}{2} \)

Step 4: Compute the limit

Now we compute the limit \( \lim_{n \to \infty} \frac{10n+10+3}{10n+3} \times \frac{1}{2} = \frac{10}{2} = 0.5 \)

Step 5: Draw conclusion

As the limit \( 0.5 < 1 \), according to the ratio test, the series converges.

Key Concepts

Ratio Test

The Ratio Test is a powerful tool used to assess the convergence of an infinite series. It involves calculating the limit of the absolute value of the ratio of successive terms in the series. Here’s how it works:

1. Identify the general term of your series, often denoted as \( a_n \). In our given exercise, the general term is \( a_n = \frac{10n+3}{n2^n} \).2. Compute the next term, \( a_{n+1} \), for the given series. 3. Form the ratio of consecutive terms \( \left| \frac{a_{n+1}}{a_n} \right| \) and determine its limit as \( n \) approaches infinity.4. Interpret the result: * If the limit \( L < 1 \), the series converges. * If \( L > 1 \) or the limit is infinite, the series diverges. * If \( L = 1 \), the test is inconclusive.
In our example, after calculating the limit for the series \( \sum_{n=1}^{\infty} \frac{10n+3}{n2^n} \), we found that \( L = 0.5 \), which is less than 1, indicating that the series converges.

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. Mathematically, it is expressed as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the general term of the sequence. Infinite series can be either convergent or divergent.

- **Convergent Series:** If the sum of an infinite series approaches a finite limit as more terms are added, then the series is said to converge. This means it approaches a specific value.
- **Divergent Series:** Conversely, if the sum grows without bound or oscillates indefinitely, it is deemed divergent.

Infinite series appear in many areas of mathematics and applied sciences, such as calculus, physics, or finance, providing a method to approach problems involving continuous growth, decay, and approximation methods.

Convergence and Divergence of Series

The convergence or divergence of a series is a fundamental concept in understanding infinite series. Essentially, these terms describe whether adding up an infinite number of terms reaches a fixed number or not.

**Tests for Convergence:** There are several techniques for determining convergence or divergence. Some commonly used tests include:

• The Ratio Test
• The Root Test
• The Comparison Test
• The Integral Test
• The Alternating Series Test

Each test has its unique conditions and applicability depending on the series being evaluated.

**Why it Matters:** Understanding whether a series converges or diverges is crucial as it impacts mathematical computations such as finding limits and solving differential equations. In the case of our exercise, by applying the ratio test, we found that the series \( \sum_{n=1}^{\infty} \frac{10n+3}{n2^n} \) converges, meaning it sums to a finite value.