Determine whether the integral is convergent or divergent.
∫[-∞,0](ze^7z) dz
since e^(7z) shrinks much faster than z grows, the integral converges.
In fact, you can evaluate it and see that the limit is -1/49
I don't understand how you get -1/49. steps please
To determine whether the integral ∫[-∞,0] (ze^7z) dz is convergent or divergent, we can use the integral convergence test.
Here's how you can find the answer step by step:
1. Start by examining the behavior of the integrand as z approaches -∞.
- When z approaches -∞, e^7z approaches zero because the exponent (7z) is a negative number.
- Additionally, z approaches -∞, so the product ze^7z approaches zero as well.
2. As we approach z = 0, the integrand z*e^7z does not present any obvious convergence issues.
3. To determine whether the integral is convergent or divergent, we need to evaluate the limit of the integrand as z approaches -∞.
lim (z -> -∞) (ze^7z) = 0
Since the limit is zero, we can conclude that the integrand approaches zero as z approaches -∞.
4. Having determined that the integrand approaches zero, we can now conclude that the integral is convergent.
Hence, the integral ∫[-∞,0] (ze^7z) dz is convergent.