Not checking the conditions of the Integral Test

Oct 13, 2025 by windailys.com 59 views

Determine the Convergence or Divergence of the Series using the Integral Test

The Integral Test is a powerful tool used in calculus to determine whether a series is convergent or divergent. It is based on the relationship between the series and its corresponding integral. In this article, we will use the Integral Test to determine the convergence or divergence of the series $\sum_{k=1}^{\infty} \frac{1}{e^k}$ .

Understanding the Integral Test

The Integral Test states that if we have a function $f(x)$ that is positive, continuous, and decreasing on the interval $[1, \infty)$ , then the series $\sum_{k=1}^{\infty} f(k)$ and the improper integral $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.

Checking the Conditions of the Integral Test

To apply the Integral Test, we need to check that the conditions are met. In this case, we have the function $f(x) = \frac{1}{e^x}$ . We need to show that this function is positive, continuous, and decreasing on the interval $[1, \infty)$ .

Positive: The function $f(x) = \frac{1}{e^x}$ is always positive for $x \geq 1$ , since $e^x > 0$ for all $x$ .
Continuous: The function $f(x) = \frac{1}{e^x}$ is continuous on the interval $[1, \infty)$ , since it is the composition of continuous functions.
Decreasing: The function $f(x) = \frac{1}{e^x}$ is decreasing on the interval $[1, \infty)$ , since the exponential function $e^x$ is increasing, and the reciprocal of an increasing function is decreasing.

Applying the Integral Test

Now that we have shown that the conditions of the Integral Test are met, we can apply the test to determine the convergence or divergence of the series $\sum_{k=1}^{\infty} \frac{1}{e^k}$ .

The corresponding integral is $\int_{1}^{\infty} \frac{1}{e^x} dx$ . To evaluate this integral, we can use the fact that the antiderivative of $\frac{1}{e^x}$ is $-\ln(e^x) = -x$ .

Therefore, we have:

\int_{1}^{\infty} \frac{1}{e^x} dx = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{e^x} dx = \lim_{b \to \infty} (-x) \Big|_{1}^{b} = \lim_{b \to \infty} (-b + 1) = -\infty

Since the integral diverges, we conclude that the series $\sum_{k=1}^{\infty} \frac{1}{e^k}$ also diverges.

Conclusion

In this article, we used the Integral Test to determine the convergence or divergence of the series $\sum_{k=1}^{\infty} \frac{1}{e^k}$ . We showed that the conditions of the Integral Test are met, and then applied the test to conclude that the series diverges.

The Integral Test is a powerful tool for determining the convergence or divergence of series, and it is an important concept in calculus. By understanding the conditions of the Integral Test and how to apply it, we can determine the convergence or divergence of a wide range of series.

References

[1] Calculus by Michael Spivak
[2] Calculus: Early Transcendentals by James Stewart
[3] Introduction to Calculus by Michael Sullivan

Further Reading

The Integral Test: A comprehensive overview of the Integral Test and its applications.
Series and Convergence: A detailed discussion of series and convergence, including the Integral Test.
Calculus: A comprehensive textbook on calculus, including the Integral Test and its applications.

Glossary

Convergence: The property of a series that it approaches a finite limit as the number of terms increases.
Divergence: The property of a series that it does not approach a finite limit as the number of terms increases.
Integral Test: A test used to determine the convergence or divergence of a series by comparing it to the corresponding integral.
Improper Integral: An integral that has an infinite limit of integration.
Series: A sum of terms, where each term is a function of the index of the term.
Q&A: The Integral Test and Series Convergence

In our previous article, we used the Integral Test to determine the convergence or divergence of the series $\sum_{k=1}^{\infty} \frac{1}{e^k}$ . In this article, we will answer some common questions about the Integral Test and series convergence.

Q: What is the Integral Test?

A: The Integral Test is a test used to determine the convergence or divergence of a series by comparing it to the corresponding integral. It states that if we have a function $f(x)$ that is positive, continuous, and decreasing on the interval $[1, \infty)$ , then the series $\sum_{k=1}^{\infty} f(k)$ and the improper integral $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.

Q: What are the conditions of the Integral Test?

A: The conditions of the Integral Test are:

Positive: The function $f(x)$ must be positive on the interval $[1, \infty)$ .
Continuous: The function $f(x)$ must be continuous on the interval $[1, \infty)$ .
Decreasing: The function $f(x)$ must be decreasing on the interval $[1, \infty)$ .

Q: How do I apply the Integral Test?

A: To apply the Integral Test, you need to:

Check that the conditions of the Integral Test are met.
Evaluate the corresponding integral.
Determine whether the integral converges or diverges.
Conclude whether the series converges or diverges based on the result of the integral.

Q: What are some common mistakes to avoid when using the Integral Test?

A: Some common mistakes to avoid when using the Integral Test include:

Not checking the conditions of the Integral Test: Make sure to check that the function is positive, continuous, and decreasing on the interval $[1, \infty)$ .
Not evaluating the corresponding integral correctly: Make sure to evaluate the integral correctly and determine whether it converges or diverges.
Not concluding correctly: Make sure to conclude correctly based on the result of the integral.

Q: Can the Integral Test be used for all series?

A: No, the Integral Test cannot be used for all series. The Integral Test is only applicable to series that meet the conditions of the Integral Test.

Q: What are some other tests for series convergence?

A: Some other tests for series convergence include:

The Ratio Test: This test is used to determine whether a series converges or diverges by comparing the ratio of consecutive terms.
The Root Test: This test is used to determine whether a series converges or diverges by comparing the nth root of the nth term.
The Comparison Test: This test is used to determine whether a series converges or diverges by comparing it to a known convergent or divergent series.

Q: How do I choose which test to use?

A: To choose which test to use, you need to consider the following factors:

The type of series: Different tests are applicable to different types of series.
The conditions of the test: Make sure to check that the conditions of the test are met.
The complexity of the test: Some tests are more complex than others, so choose a test that is suitable for your level of expertise.

Conclusion

In this article, we answered some common questions about the Integral Test and series convergence. We hope that this article has been helpful in understanding the Integral Test and how to apply it. If you have any further questions, please don't hesitate to ask.

References

[1] Calculus by Michael Spivak
[2] Calculus: Early Transcendentals by James Stewart
[3] Introduction to Calculus by Michael Sullivan

Further Reading

The Integral Test: A comprehensive overview of the Integral Test and its applications.
Series and Convergence: A detailed discussion of series and convergence, including the Integral Test.
Calculus: A comprehensive textbook on calculus, including the Integral Test and its applications.

Glossary

Convergence: The property of a series that it approaches a finite limit as the number of terms increases.
Divergence: The property of a series that it does not approach a finite limit as the number of terms increases.
Integral Test: A test used to determine the convergence or divergence of a series by comparing it to the corresponding integral.
Improper Integral: An integral that has an infinite limit of integration.
Series: A sum of terms, where each term is a function of the index of the term.