Problem 10 - Thomas Calculus

Question

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=2}^{\infty} \frac{n-4}{n^{2}-2 n+1} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Simplify the Series Terms

First, simplify the term \( \frac{n-4}{n^{2}-2 n+1} \). Notice that the denominator can be rewritten as \( (n-1)^2 \). Thus, the series becomes:\[\sum_{n=2}^{\infty} \frac{n-4}{(n-1)^2}\]

Step 2: Verify the Conditions for the Integral Test

The Integral Test requires that the function \( f(x) = \frac{x-4}{(x-1)^2} \) is positive, continuous, and decreasing for \( x \geq 2 \). - **Positive:** For \( x \geq 2 \), the numerator \(x-4\) is negative, making \(f(x) > 0 \) untrue until further analyzed.- **Continuous:** The function is continuous for \( x eq 1 \).- **Decreasing:** Evaluate \(f'(x)\) to check if it's negative for x ≥ 2.

Step 3: Analyze Function Behavior

Notice the negative behavior arises because \(x - 4\) for \(x \geq 2\) doesn’t typically meet required conditions of positivity. Verify decreasing nature by calculating derivative:\[f'(x) = \frac{(x-1)^2 - (x-4)\cdot(2(x-1))}{((x-1)^2)^2} = \frac{(x-1)^2 - 2(x-4)(x-1)}{(x-1)^4}\] Simplify to check if \(f(x)\) is decreasing for \(x \geq 2\).

Step 4: Apply the Integral Test

Assuming earlier step shows valid decreasing nature, evaluate:\[\int_{2}^{\infty} \frac{x-4}{(x-1)^2} \, dx\]To evaluate the integral, start by making the substitution \( u = x - 1 \), \( du = dx \), transforming limits appropriately.

Step 5: Evaluate the Integral

With substitution \( u = x -1 \), \( x = u + 1 \):\[\int_{1}^{\infty} \frac{u-3}{u^2} \, du = \int_{1}^{\infty} \left(\frac{1}{u} - \frac{3}{u^2}\right) \, du \]Evaluate each integral term separately to find their convergence/divergence.\[\int \frac{1}{u} \, du = \ln|u|\]\[\int \frac{3}{u^2} \, du = -\frac{3}{u}\]Evaluate improper limits as necessary.

Step 6: Determine Convergence or Divergence

Each term simplifies to:\[[\ln|u|]_{1}^{\infty} - [-3/u]_{1}^{\infty}\]The natural log diverges to infinity, implying the entire integral diverges.

Key Concepts

Series Convergence

When dealing with infinite series, one of the main questions is whether the series converges or diverges. A series converges if the sum of its terms approaches a finite value as more and more terms are added. Conversely, it diverges if it fails to approach a finite value. Understanding this is crucial because it determines whether the sum is meaningful or not.
To determine convergence, different tests can be applied. The Integral Test is one method that relates convergence of a series to the convergence of an improper integral. It's worth noting that the series sum often relies on the behavior of terms as the index gets very large, typically infinity.

• Convergent series means it sums to a finite number.
• Divergent series means it does not sum to a finite number.
• The Integral Test can be a key tool in determining which case a given series falls into.

In practical applications, series convergence can be essential in data computation, physics problem-solving, and mathematical financial models.

Improper Integrals

Improper integrals extend the concept of integration to functions that have infinite limits or exhibit behaviors like vertical asymptotes. In the context of testing series convergence through the Integral Test, evaluating improper integrals becomes crucial.
For example, if we examine \[\int_{2}^{\infty} \frac{x-4}{(x-1)^2} \, dx\]in the original exercise, the upper limit reaches infinity, making it an improper integral. The convergence of this integral helps determine the convergence of an associated series. Here are key points to remember:

• Check if the function has any discontinuities. Such as vertical asymptotes within the interval of integration.
• The substitution method can simplify evaluation. Often used to rewrite terms into a solvable form.
• Assess each part of the integral separately if decomposing is necessary.

Through these practices, improper integrals can be evaluated correctly, providing insight into the behavior and sum of infinite series.

Convergence Tests

Convergence tests are numerous, each with its own application domain and utility. The Integral Test is one of the common tests used for determining series convergence, but others like the Ratio Test, Root Test, and Comparison Test exist as well. Each test comes with specific criteria for when it can be applied.
The Integral Test requires you to find a continuous, positive, decreasing function that approximates the series terms for sufficiently large indices. Once these conditions are verified, the convergence of the corresponding improper integral will determine the series' convergence.

• Integral Test: Links series convergence to integrals.
• Ratio Test: Compares ratios of successive terms.
• Comparison Test: Involves comparing with known series.

Understanding which test to apply often depends on the function's characteristics. Always verify that necessary conditions of the chosen test match before proceeding to ensure accurate results. This can save time and prevent errors in determining series convergence.