Integral of x^n e^(-x) | MIT 18.01SC Single Variable Calculus, Fall 2010
TLDRIn this educational video, the presenter guides viewers through a mathematical proof to demonstrate the convergence of the integral of x^n * e^(-x) from 1 to infinity for any value of n, without using integration by parts repeatedly. The strategy involves comparing the given function to a modified version, e^(x/2) * e^(-x), which simplifies to a function that is known to converge. The presenter illustrates the thought process and the importance of choosing the right comparison function to establish convergence, ultimately showing that the integral in question converges to a finite value.
Takeaways
- 📚 The video aims to demonstrate the convergence of the integral of x^n * e^(-x) from 1 to infinity for any value of n.
- 🔍 The goal is to show convergence without using integration by parts n times.
- 📉 The concept of convergence means that the integral's value approaches a finite number as the limit of the upper bound goes to infinity.
- 🤔 The initial thought is to compare x^n to e^x, but it's realized that this comparison alone isn't sufficient to prove convergence.
- 🔄 A comparison with e^x is attempted but leads to an integral of 1, which diverges, indicating a different approach is needed.
- 💡 The strategy involves finding a constant multiplier of e^x that, when compared to x^n, will ensure the integral converges.
- 🔢 Choosing x^(2n) as a comparison to e^x is suggested, as it is smaller for large enough x.
- 📈 The integral of e^(x/2) * e^(-x) is considered, which simplifies to e^(-x/2), a function that decays rapidly to zero.
- 📝 An antiderivative of e^(-x/2) is found, which confirms the convergence of the integral from R to infinity.
- 📌 The integral from 1 to R of the original function is finite, and from R to infinity, it converges, thus proving the overall convergence.
- 🧩 The process involves a combination of comparison, substitution, and understanding the behavior of exponential functions to establish convergence.
Q & A
What is the main objective of the video?
-The main objective of the video is to demonstrate that the integral of x to the power of n multiplied by e to the power of negative x, from 1 to infinity, converges for any value of n, without using integration by parts n times.
What does it mean for an integral to converge?
-An integral converges if, for a fixed value b, the limit as b approaches infinity of the sequence of values converges to a finite number. In other words, the area under the curve defined by the function over the given interval does not approach infinity.
Why is it sufficient to show the integral converges from a very large number to infinity?
-It is sufficient because the function is continuous from 1 to infinity, ensuring that there are no infinite areas in finite intervals. If the integral from a large number R to infinity converges, then the integral from 1 to R will also be finite, and thus the entire integral from 1 to infinity will converge.
What is the initial comparison that was considered but deemed insufficient?
-The initial comparison considered was between x to the power of n and e to the power of x. It was thought that since x to the n is much smaller than e to the x for large x, the integral of the former might be smaller than the integral of the latter. However, this comparison was insufficient because the integral of e to the x times e to the minus x diverges.
Why does the integral of e to the power of x times e to the power of negative x diverge?
-The integral diverges because e to the power of x times e to the power of negative x simplifies to e to the power of 0, which is 1. Integrating the constant 1 from a large number R to infinity results in an infinite area.
What strategy is used to show the convergence of the integral?
-The strategy involves comparing the original function to a modified function where x is raised to the power of 2n instead of n, and then comparing it to e to the power of x over 2. This comparison shows that for sufficiently large x, x to the 2n is much less than e to the x, allowing the integral to converge.
Why is e to the power of x over 2 a better comparison than e to the power of x?
-E to the power of x over 2 is a better comparison because when you multiply it by e to the power of negative x, you get e to the power of negative x over 2, which decays rapidly as x approaches infinity, ensuring the integral converges.
How does the讲师 handle the comparison to show the integral converges?
-The讲师 compares the integral of x to the power of n times e to the power of negative x to the integral of e to the power of x over 2 times e to the power of negative x. The latter integral is shown to converge, and since the original integral is less than this, it also converges.
What is the antiderivative of e to the power of negative x over 2?
-The antiderivative is -2 times e to the power of negative x over 2, which can be derived by using a substitution method or recognizing the pattern of the exponential function's integral.
How does the讲师 conclude the integral from 1 to infinity converges?
-The讲师 concludes by showing that the integral from a large number R to infinity of the modified function converges, and since the integral from 1 to R is finite, the entire integral from 1 to infinity must also converge.
What is the significance of the讲师's approach in this video?
-The significance of the讲师's approach is to demonstrate a method for proving the convergence of an integral without resorting to repeated integration by parts, which can be cumbersome and less intuitive for understanding the underlying concepts.
Outlines
📚 Demonstrating Convergence of an Integral
The video script introduces a mathematical problem where the goal is to prove the convergence of the integral of x^n * e^(-x) from 1 to infinity, for any value of n, without using integration by parts multiple times. The instructor emphasizes the definition of convergence in the context of integrals and hints at a strategy to show the integral converges by comparing it to a simpler function.
🔍 Exploring the Strategy for Proving Convergence
The script delves into a strategy for proving the convergence of the integral by comparing it to the integral of e^(x/2) * e^(-x) from a large value R to infinity, which is shown to converge. The instructor explains the thought process behind choosing a comparison function that decays faster than x^n and demonstrates the calculation of the antiderivative of e^(-x/2), leading to the conclusion that the original integral must also converge.
📈 Concluding the Proof of Convergence
The final paragraph wraps up the proof by reiterating the strategy used and summarizing the key points. It clarifies that the convergence of the integral from R to infinity of the comparison function implies the convergence of the original integral. The instructor also explains that the integral from 1 to R is finite, thus confirming the convergence of the integral from 1 to infinity of the original function.
Mindmap
Keywords
💡Recitation
💡Integral
💡Convergence
💡Integration by parts
💡Exponential function
💡Comparison test
💡Diverges
💡Antiderivative
💡Continuous function
💡Power of x
Highlights
The video aims to demonstrate the convergence of the integral of x^n * e^(-x) from 1 to infinity for any value of n, without using integration by parts.
Convergence is defined as the limit of the integral values converging to a finite number as the upper limit approaches infinity.
The integral's convergence is to be shown without repeatedly using integration by parts to eliminate powers of x.
A strategy is proposed to show convergence by comparing the integral from a large value R to infinity, rather than from 1 to infinity.
The idea that x^n is much smaller than e^x for large x is introduced as a potential method for comparison.
The comparison of x^n to e^x leads to an integral that diverges, indicating a need for a different approach.
A modified approach is suggested by comparing x^n to e^(x/2) instead of e^x to find a convergent integral.
The selection of the constant 2 in x^(2n) is explained as a strategic choice for the comparison.
A formal argument is constructed to show that x^n is much smaller than e^(x/2) for x greater than a certain R.
The integral of e^(x/2) * e^(-x) is simplified to show its convergence, which is key to the argument.
The antiderivative of e^(-x/2) is calculated to demonstrate the convergence of the integral.
The strategy of comparing the original integral to a known convergent integral is explained.
The importance of the fast decay of e^(-x) in achieving convergence is highlighted.
An alternative comparison to x^(-2) is mentioned to show the rapid decay of the function.
The conclusion that the original integral converges is reached by comparing it to the convergent integral of e^(-x/2).
The process of elimination and refinement of the comparison function is outlined as a problem-solving strategy.
The final step confirms the convergence of the original integral from 1 to infinity by leveraging the convergence from R to infinity.
The video concludes with a summary of the thought process and strategy used to demonstrate the integral's convergence.
Transcripts
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