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
is convergent or divergent using the Integral Test, we evaluate the improper integral
Step 1: Set up the Integral
The function we need to integrate is .
Step 2: Use a Substitution
Let . Then, , or equivalently .
When , . When , .
Substituting into the integral, we get:
Step 3: Evaluate the Integral
Now we have:
The integral of is , so we get:
Conclusion
Since the integral converges to a finite value, by the Integral Test, the series is convergent.
Would you like further details on each step or have any questions?
Follow-Up Questions:
- How does the Integral Test work, and why does it imply convergence?
- What other convergence tests could we use for this series?
- Could a comparison test be applied to this problem?
- Why is the substitution helpful in evaluating this integral?
- How does this series compare to a simpler exponential series?
Tip:
When using the Integral Test, check that the function is positive, continuous, and decreasing for to apply the test correctly.
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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