https://file.aiquickdraw.com/mathbot/file/1730136446039beo485x7.jpg

To determine whether the series 

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

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

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

### Step 1: Set up the Integral
The function we need to integrate is \( f(x) = x^3 e^{-x^4} \).

### Step 2: Use a Substitution
Let \( u = x^4 \). Then, \( \frac{du}{dx} = 4x^3 \), or equivalently \( dx = \frac{du}{4x^3} \).

When \( x = 1 \), \( u = 1^4 = 1 \).
When \( x \to \infty \), \( u \to \infty \).

Substituting into the integral, we get:

\[
\int_{1}^{\infty} x^3 e^{-x^4} \, dx = \int_{1}^{\infty} e^{-u} \cdot \frac{du}{4} = \frac{1}{4} \int_{1}^{\infty} e^{-u} \, du.
\]

### Step 3: Evaluate the Integral
Now we have:

\[
\frac{1}{4} \int_{1}^{\infty} e^{-u} \, du.
\]

The integral of \( e^{-u} \) is \( -e^{-u} \), so we get:

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

### Conclusion
Since the integral converges to a finite value, by the Integral Test, the series \( \sum_{n=1}^{\infty} n^3 e^{-n^4} \) 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 \( u = x^4 \) 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 \( f(x) \) is positive, continuous, and decreasing for \( x \geq 1 \) to apply the test correctly.

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.

Learn how to use the Integral Test to determine if a series is convergent by evaluating the improper integral ∫[1,∞] x³e^(-x⁴) dx. This example covers essential calculus concepts suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation