Infinite series to calculate integrals

In summary, the conversation discusses issues with a specific integral and the study of power series in complex analysis. The issues of convergence, derivatives, and integrals are addressed in detail in the study of power series. The speaker also mentions the use of series for calculating definite integrals, but notes that it may not always provide accurate results.
  • #1
fazekasgergely
3
0
Homework Statement
I've been doing maths for fun (I've only learned calculus 1 in university yet, but I love experimenting) and I've found the formula for integration by parts can be generalized as an infinite series.
Relevant Equations
I'm not well educated in infinite series, this is an amateur attempt but it works in most cases. Can someone tell me if this is right? I know in most cases it only makes things more complicated. I've tried it with simpler functions and it seems to work, more complicated ones are sometimes a problem. Here d^0/dx^0 f(x)=f(x)
For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
 

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  • #2
The study of power series (Taylor series) is central to the subject of complex analysis. The issues of convergence, derivatives, and integrals are handled in detail there. You might be interested in that subject.
Your equation seems to have some problems. For one, the variable x inside the integral on the left is a dummy variable, but not on the right. Also, consider the series for the exponential function. The integral is the exponential function again and it does not have the alternating sign that your equation has.
 
  • #3
Thank you for your answer! It's good to read the opinion of someone who is more educated than myself. For now I'm just an enthusiast who enjoys experimenting with maths :)
 
  • #4
I've found if you plug in e^x the series gives you e^x-1, and by definition integral of e^x is e^x+C, so the derivative of e^x-1 is still e^x. But the series is not great for calculating definite integral, the results will be false. It works for some functions though.
 

1. What is an infinite series?

An infinite series is a mathematical expression that represents the sum of an infinite number of terms. It is written in the form of a sum, where each term is added to the previous one.

2. How can infinite series be used to calculate integrals?

Infinite series can be used to approximate the value of an integral by breaking it down into smaller, simpler parts. By using a specific type of infinite series, called a Taylor series, we can represent a function as an infinite sum of polynomials. This allows us to estimate the value of the integral by adding up these polynomials.

3. What is the difference between a convergent and divergent infinite series?

A convergent infinite series is one in which the sum of all the terms approaches a finite value as the number of terms increases. This means that the series has a definite value. On the other hand, a divergent infinite series is one in which the sum of the terms does not approach a finite value, and therefore the series has no definite value.

4. How do you determine the convergence of an infinite series?

There are several tests that can be used to determine the convergence of an infinite series, such as the comparison test, ratio test, and integral test. These tests involve comparing the given series to a known convergent or divergent series or evaluating the limit of the series' terms. If the limit is equal to zero, the series is convergent, and if the limit is non-zero, the series is divergent.

5. Are there any limitations to using infinite series to calculate integrals?

Yes, there are limitations to using infinite series to calculate integrals. One limitation is that the series may not converge for certain functions or values of x, making it impossible to accurately calculate the integral. Additionally, using infinite series to calculate integrals can be time-consuming and may not always provide an exact solution, only an approximation.

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