To prove that the limits $I_n = \lim_{T \to \infty} \int_0^T t^n e^{-t} \, dt$ exist and to find their values, we can proceed step by step.

Step 1: Base Case

Start with $n = 0$. The integral becomes: $I_0 = \lim_{T \to \infty} \int_0^T e^{-t} \, dt$ This is a straightforward integral: $\int_0^T e^{-t} \, dt = \left[-e^{-t}\right]_0^T = 1 - e^{-T}.$ As $T \to \infty$, $e^{-T} \to 0$, so: $I_0 = 1.$ Thus, the base case holds: $I_0 = 1$.

Step 2: Inductive Step

Now, assume that for some $n = k$, the following is true: $I_k = \lim_{T \to \infty} \int_0^T t^k e^{-t} \, dt \quad \text{exists and is finite}.$ We want to show that the integral for $n = k+1$ also exists and find its value.

Integral for $n = k+1$

Consider the integral: $I_{k+1} = \lim_{T \to \infty} \int_0^T t^{k+1} e^{-t} \, dt.$ We can integrate by parts, where we let:

• $u = t^{k+1}$, so that $du = (k+1)t^k \, dt$,
• $dv = e^{-t} \, dt$, so that $v = -e^{-t}$.

Using the integration by parts formula: $\int u \, dv = uv - \int v \, du,$ we have: $\int_0^T t^{k+1} e^{-t} \, dt = \left[ -t^{k+1} e^{-t} \right]_0^T + (k+1) \int_0^T t^k e^{-t} \, dt.$ Now, evaluate the boundary terms: $\left[ -t^{k+1} e^{-t} \right]_0^T = -T^{k+1} e^{-T} + 0 = -T^{k+1} e^{-T}.$ As $T \to \infty$, $T^{k+1} e^{-T} \to 0$ because the exponential decay of $e^{-T}$ dominates any polynomial growth of $T^{k+1}$. So, the boundary term vanishes: $\lim_{T \to \infty} -T^{k+1} e^{-T} = 0.$ Thus, we are left with: $I_{k+1} = (k+1) \lim_{T \to \infty} \int_0^T t^k e^{-t} \, dt = (k+1) I_k.$

Step 3: Conclusion

By induction, we have shown that $I_n = n!$ for all $n$. Specifically:

• $I_0 = 1 = 0!$,
• $I_1 = 1 = 1!$,
• $I_2 = 2 = 2!$,
• and so on.

Therefore, we conclude: $I_n = \lim_{T \to \infty} \int_0^T t^n e^{-t} \, dt = n! \quad \text{for all} \quad n = 0, 1, 2, \dots$

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Would you like a more detailed explanation of any of the steps or further examples? Here are 5 related questions you might find interesting:

1. How does the method of integration by parts generalize for higher-order integrals?
2. How does the exponential decay $e^{-t}$ affect the convergence of integrals as $T \to \infty$?
3. What is the relationship between the Gamma function and factorials in terms of integrals?
4. Can we derive the factorial expression using a recurrence relation instead of induction?
5. How does the integral $\int_0^\infty t^n e^{-t} \, dt$ relate to the Gamma function?

Tip: Whenever you're dealing with limits of integrals involving exponential decay, check if integration by parts can simplify the process and eliminate troublesome terms.