Math Problem Statement

Use the Integral Test to determine whether the series is convergent or divergent. Evaluate the following integral: ∫[1,∞] x³e^(-x⁴) dx. Since the integral is finite, the series is convergent.

Solution

To determine whether the series

n=1n3en4\sum_{n=1}^{\infty} n^3 e^{-n^4}

is convergent or divergent using the Integral Test, we evaluate the improper integral

1x3ex4dx.\int_{1}^{\infty} x^3 e^{-x^4} \, dx.

Step 1: Set up the Integral

The function we need to integrate is f(x)=x3ex4f(x) = x^3 e^{-x^4}.

Step 2: Use a Substitution

Let u=x4u = x^4. Then, dudx=4x3\frac{du}{dx} = 4x^3, or equivalently dx=du4x3dx = \frac{du}{4x^3}.

When x=1x = 1, u=14=1u = 1^4 = 1. When xx \to \infty, uu \to \infty.

Substituting into the integral, we get:

1x3ex4dx=1eudu4=141eudu.\int_{1}^{\infty} x^3 e^{-x^4} \, dx = \int_{1}^{\infty} e^{-u} \cdot \frac{du}{4} = \frac{1}{4} \int_{1}^{\infty} e^{-u} \, du.

Step 3: Evaluate the Integral

Now we have:

141eudu.\frac{1}{4} \int_{1}^{\infty} e^{-u} \, du.

The integral of eue^{-u} is eu-e^{-u}, so we get:

14[eu]1=14(0+e1)=14e.\frac{1}{4} \left[ -e^{-u} \right]_{1}^{\infty} = \frac{1}{4} \left( 0 + e^{-1} \right) = \frac{1}{4e}.

Conclusion

Since the integral converges to a finite value, by the Integral Test, the series n=1n3en4\sum_{n=1}^{\infty} n^3 e^{-n^4} is convergent.

Would you like further details on each step or have any questions?

Follow-Up Questions:

  1. How does the Integral Test work, and why does it imply convergence?
  2. What other convergence tests could we use for this series?
  3. Could a comparison test be applied to this problem?
  4. Why is the substitution u=x4u = x^4 helpful in evaluating this integral?
  5. How does this series compare to a simpler exponential series?

Tip:

When using the Integral Test, check that the function f(x)f(x) is positive, continuous, and decreasing for x1x \geq 1 to apply the test correctly.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Series Convergence
Improper Integrals

Formulas

Integral Test for convergence
∫[1,∞] x³e^(-x⁴) dx

Theorems

Integral Test

Suitable Grade Level

Grades 11-12