Confirm That The Integral Test Can Be Applied To The Series. Then Use The Integral Test To Determine
Understanding the convergence or divergence of infinite series is a fundamental topic in calculus, especially in the study of series. One of the most powerful tools available for this purpose is the Integral Test. Before applying the Integral Test, it is crucial to confirm that the series in question meets the necessary conditions for the test's validity. This article provides a comprehensive guide to confirming the applicability of the Integral Test, explaining the criteria involved, and demonstrates how to use the test to analyze the convergence of a series with detailed examples.
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What Is the Integral Test?
The Integral Test is a method used to determine whether an infinite series converges or diverges by comparing it to a related improper integral. Specifically, it applies to series whose terms are positive, decreasing, and continuous functions over a certain interval.
Formal Statement of the Integral Test:
Suppose \(f(n) = a_n\) is a continuous, positive, decreasing function for \(x \geq N\), where \(N\) is a positive integer. Then, the series
\[
\sum{n=N}^{\infty} an
\]
and the improper integral
\[
\int_{N}^{\infty} f(x) \, dx
\]
either both converge or both diverge.
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Conditions for Applying the Integral Test
Before applying the Integral Test, verify that the series satisfies the following key conditions:
1. Positivity of the Terms
- All terms \(an\) in the series must be positive (\(an > 0\)) for sufficiently large \(n\).
- This ensures the integral comparison is meaningful because the integral of a positive function provides an accurate basis for convergence analysis.
2. Monotonic Decrease of the Function
- The function \(f(x) = a_x\) must be decreasing for all \(x \geq N\).
- That is, \(f'(x) \leq 0\) for \(x \geq N\).
- Monotonicity guarantees that the integral comparison is valid because the function does not oscillate.
3. Continuity of the Function
- The function \(f(x)\) should be continuous on the interval \([N, \infty)\).
- Continuity ensures that the integral is well-defined and can be evaluated or estimated accurately.
4. Proper Behavior at Infinity
- The improper integral \(\int_{N}^{\infty} f(x) \, dx\) should be properly defined.
- For convergence, the improper integral must approach a finite value as the upper limit tends to infinity.
Steps to Confirm the Applicability of the Integral Test
To systematically confirm whether the Integral Test can be applied, follow these steps:
- Identify the general term \(a_n\): Extract the expression for the series term.
- Define the corresponding function \(f(x)\): Replace \(n\) with \(x\) to get a continuous function \(f(x)\).
- Verify positivity: Confirm \(f(x) > 0\) for all \(x \geq N\).
- Check for monotonicity: Determine whether \(f(x)\) is decreasing for \(x \geq N\). You can do this by computing \(f'(x)\).
- Ensure continuity: Confirm \(f(x)\) is continuous on \([N, \infty)\).
- Identify \(N\): Choose an appropriate \(N\) such that all conditions hold for \(x \geq N\).
Once these conditions are satisfied, the Integral Test can be confidently applied to analyze the series.
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Applying the Integral Test: Step-by-Step Example
Let's apply this process to a concrete example to illustrate how to confirm the conditions and then determine the series' convergence.
Example Series:
\[
\sum_{n=2}^{\infty} \frac{1}{n^p}
\]
where \(p\) is a real number.
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Step 1: Identify \(a_n\)
\[
a_n = \frac{1}{n^p}
\]
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Step 2: Define \(f(x)\)
\[
f(x) = \frac{1}{x^p}
\]
Note: \(f(x)\) is defined for \(x \geq 1\) (or \(x \geq 2\) to match the series).
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Step 3: Verify positivity
- For \(x > 0\), \(f(x) = \frac{1}{x^p} > 0\) as long as \(x \neq 0\).
- Since \(x \geq 2\), the function is positive.
Step 4: Check for monotonicity
- Compute the derivative:
- For \(x \geq 2\):
- If \(p > 0\), then \(f'(x) < 0\), indicating the function is decreasing.
- If \(p \leq 0\), the function may not be decreasing; further analysis needed.
- Conclusion: For \(p > 0\), \(f(x)\) is decreasing on \([2, \infty)\).
Step 5: Confirm continuity
- \(f(x)\) is continuous for \(x > 0\).
Step 6: Choose \(N\)
- Since the conditions hold for \(x \geq 2\), set \(N=2\).
Using the Integral Test to Determine Convergence
Having confirmed the conditions, now evaluate the improper integral:
\[
\int_{2}^{\infty} \frac{1}{x^{p}} \, dx
\]
Evaluation:
- For \(p \neq 1\):
\[
\int{2}^{\infty} x^{-p} \, dx = \left[ \frac{x^{-p+1}}{-p+1} \right]2^{\infty}
\]
- Case 1: \(p > 1\)
- \(-p + 1 < 0\), so as \(x \to \infty\), \(x^{-p+1} \to 0\).
- Therefore,
\[
\int_{2}^{\infty} \frac{1}{x^{p}} \, dx = \frac{2^{-p+1}}{p-1}
\]
which is finite.
- Case 2: \(p \leq 1\)
- When \(p = 1\):
\[
\int{2}^{\infty} \frac{1}{x} \, dx = \lim{t \to \infty} \ln t - \ln 2 = \infty
\]
diverges to infinity.
- When \(p < 1\):
\[
\int{2}^{\infty} x^{-p} \, dx = \lim{t \to \infty} \frac{t^{-p+1}}{-p+1}
\]
Since \(-p + 1 > 0\), as \(t \to \infty\), \(t^{-p+1} \to \infty\), so the integral diverges.
Summary of Results:
| \(p\) Value | Series Convergence | Explanation |
|--------------|---------------------|-------------|
| \(p > 1\) | Converges | The integral converges to a finite value. |
| \(p \leq 1\) | Diverges | The integral diverges to infinity. |
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Conclusion: Series Convergence Based on the Integral Test
- The series \(\sum_{n=2}^{\infty} \frac{1}{n^p}\) converges if and only if \(p > 1\).
- It diverges for \(p \leq 1\).
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Additional Tips for Applying the Integral Test
- Compare with known p-series: Many series with terms involving powers of \(n\) are directly comparable to p-series, which have well-understood convergence properties.
- Use substitution for complex integrals: When integrals involve more complicated functions, substitution or partial fractions can simplify evaluation.
- Be cautious with non-monotonic functions: The Integral Test requires the function to be decreasing; if not, consider other tests like the Comparison or Limit Comparison Test.
Summary
Applying the Integral Test effectively hinges on verifying key conditions: positivity
