Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to ∞ of (dθ / (1 + e^θ))

Accepted Answer

Identify the integral to be tested for convergence: $$\displaystyle \$$int_0$$^{\infty} \frac{d	heta}{1 + e$$^{	heta}$$}. $$Examine the behavior of the integrand $$f(	heta) = \frac{1}{1 + e$$^{	heta}$$} as$$ $$	heta 	o \infty$$ and as $$	heta 	o 0 to $$understand the nature of the integral over the interval $$[0, \infty). $$For large $$	heta$$, note that $$e^{	heta}$$ grows very quickly, so $$f(	heta)$$ behaves like $$\frac{1}{e$$^{	heta}$$} = e$$^{-	heta}$$. This suggests comparing the integrand to g$$(	heta) = $$e^{-	heta}$$, a function with a known convergent integral over $$[0, \infty). $$Apply the Direct Comparison Test by checking if $$0 \leq f(	heta) \leq g(	heta)$$ for sufficiently large $$	heta. $$Since $$\frac{1}{1 + e$$^{	heta}$$}  0$$, the integral of $$f(	heta) is $$bounded above by the integral of $$g(	heta)$$, which converges. Conclude that since $$\$$int_0$$^{\infty} e$$^{-	heta}$$ d	heta$$ converges and $$f(	heta) is $$less than or equal to $$g(	heta)$$ for large $$	heta$$, by the Direct Comparison Test, the original integral $$\$$int_0$$^{\infty} \frac{d	heta}{1 + e$$^{	heta}$$}$$ also converges.

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to ∞ of (dθ / (1 + e^θ))

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Improper Integrals

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.∫ from 0 to ∞ of (dθ / (1 + e^θ))

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Improper Integrals

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to ∞ of (dθ / (1 + e^θ))