Limit of Summation: Solving with Integration

In summary, the conversation discusses solving a problem involving a limit and summation by using integration. The final solution is expressed as the integral of a function with limits of integration from a to b.
  • #1
azatkgz
186
0

Homework Statement


[tex]\frac{1}{n}\lim_{n\rightarrow\infty}\sum_{k=1}^{n}f(a+\frac{b-a}{n}k)[/tex]





The Attempt at a Solution


I tried to solve it simply.
[tex]\frac{1}{n}\lim_{n\rightarrow\infty}\sum_{k=1}^{n}f(a+\frac{b-a}{n}k)=\int_{0}^{1}f(a+(b-a)x)dx[/tex]

[tex]=f(b)-f(a)[/tex]
 
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  • #2
Sorry,
[tex]\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}f(a+\frac{b-a}{n}k)=\int_{0}^{1}f(a+(b-a)x)dx[/tex]
 
  • #3
Very good. Now integrate by letting u= a+ (b-a)x. What is du? Also, be careful about the limits of integration.
 
  • #4
[tex]\int_{a}^{b}\frac{f(u)du}{(b-a)}=\frac{1}{b-a}(f(b)-f(a))[/tex]
Yes?
 
  • #5
Yes, to the left hand side. No, to the right. Unless the 'f' on the right stands for the antiderivative of the 'f' on the left. In which case you should say so and use a different symbol.
 
  • #6
Than ,actually,answer remains just
[tex]\frac{1}{b-a}\int_{a}^{b}f(u)du[/tex]
 
  • #7
Sure the summation is just a definition of a riemann integral.
 

What is the limit of a summation?

The limit of a summation, also known as the infinite sum or series, is the value that the sum approaches as the number of terms increases towards infinity.

How do you calculate the limit of a summation?

The limit of a summation can be calculated by first finding the general term of the series, then taking the limit of that term as the number of terms approaches infinity. This can be done using various mathematical techniques such as the ratio test or the root test.

What is the significance of the limit of a summation?

The limit of a summation is important in mathematics as it allows us to determine the behavior of a series as the number of terms increases towards infinity. It helps us understand whether a series converges or diverges, and can also be used to evaluate complicated mathematical expressions.

Can the limit of a summation be negative?

Yes, the limit of a summation can be negative. This occurs when the terms of the series alternate between positive and negative values, resulting in a net negative value as the number of terms increases towards infinity.

What are some real-world applications of the limit of a summation?

The limit of a summation is used in various fields, such as physics, engineering, and economics. For example, it can be used to calculate the total distance traveled by an object in motion, or to determine the total cost of a loan with compound interest over time.

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