Can the Maclaurin Series for Sin(x) be Integrated?

In summary: However, you can integrate the terms within the sigma notation, which in this case is the Maclaurin series for sin(x).In summary, the person was having trouble integrating the Maclaurin series for sin(x) and was unsure if it was integrable. They also mentioned trying various integration methods such as u-substitution, but were unable to find a solution. The other person explained that since the series is equal to sin(x), it is indeed integrable and can be integrated term by term. They also mentioned that the person's expertise in other areas suggests that they should have no trouble integrating the series.
  • #1
hawaiidude
41
0
i was stumbled on a problem was i was looking over my book...i can do things like, differntial geometry, Fourier analsysis, advanced calculus, calculus etc...but i can;t figure out this problem...i don't think i can figure it out...it's from the maclariun series...as most of you know, sinx=x-x^3/3!+x^5/5!-x^7/7!+..=sigma ^infinity, n=0 (-1)^nx^2n+1/ (2n+1)!...at sin(.4)...i don;t think this is integratable...i tried everything from integration to u-substituion but it won't work...
 
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  • #2
Would it be too much trouble to tell us what the problem is?

You say "it's from the maclariun series...as most of you know, sinx=x-x^3/3!+x^5/5!-x^7/7!+..=sigma ^infinity, n=0 (-1)^nx^2n+1/ (2n+1)!...at sin(.4)...i don;t think this is integratable" so I take it you want to integrate? Integrate what? Integrate sin(x) or its Maclaurin series? What do you mean "at sin(.4)"?

Certainly sin(x) is integrable. Since the Maclaurin series for sin(x) is equal to sin(x), the Maclaurin series is. What exactly is it that you want to do?
 
  • #3
integrate the sigma part
 
  • #4
"integrate the sigma part"

That was an answer? "The sigma part" of what?

Assuming that you mean "integrate the Maclaurin series for sin(x)", the point I made before was that since the series is equal to sin(x) so it certainly is integrable. Its integral is the same as the integral of sin(x): cos(x).

You can also, of course, do it term by term: the Mclaurin series for sin(x) is, as you say x- (1/3!)x3+...+ (1/(2n+1)!)x2n+1+ ... Integrating term by term gives
(1/2)x2- (1/4!)x4+ ...+ (1/(2n+2)!)x2n+2+ ... which is, of course, the Maclaurin series for cos(x).

If you can do things like "differntial geometry, Fourier analsysis, advanced calculus", I don't see why you would think sin(x) was not integrable.
 
  • #5
no was talking about the sigma notation part but o well ok
 
  • #6
you can integrate the sigma notation no more than you can integrate the + symbol.
 

What is an integration maclaurin series?

An integration maclaurin series is a mathematical series that approximates the value of a definite integral. It is derived from the Maclaurin series, which is a special case of the Taylor series. Integration maclaurin series are useful for solving integrals that cannot be solved using traditional methods.

How is an integration maclaurin series derived?

An integration maclaurin series is derived by integrating the Maclaurin series term by term. This means that each term in the Maclaurin series is integrated separately, resulting in a new series. The integration process continues until the desired level of precision is achieved.

What is the purpose of an integration maclaurin series?

The main purpose of an integration maclaurin series is to approximate the value of a definite integral. This can be useful in situations where the integral cannot be solved using traditional methods, or when a more precise value is needed.

What are the limitations of an integration maclaurin series?

An integration maclaurin series has some limitations, including the fact that it can only approximate the value of a definite integral and not provide an exact solution. It also has a limited range of convergence, meaning that it may not work for all types of integrals. Additionally, the series may require a large number of terms to achieve a desired level of precision.

What are some real-world applications of an integration maclaurin series?

Integration maclaurin series have a wide range of real-world applications, including in physics, engineering, and economics. They can be used to calculate areas and volumes of irregular shapes, approximate solutions to differential equations, and analyze complex systems. In economics, they can be used to model and predict economic variables such as inflation and interest rates.

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