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

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Discussion Overview

The discussion revolves around the integration of the Maclaurin series for sin(x), specifically whether it is possible to integrate the series represented in sigma notation. Participants explore the implications of integrating the series versus integrating the function itself.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant expresses confusion about integrating the Maclaurin series for sin(x) and doubts its integrability.
  • Another participant clarifies that since the Maclaurin series is equal to sin(x), it is integrable, and its integral is the same as that of sin(x), which is cos(x).
  • A later reply suggests that integrating the series term by term is possible, leading to the Maclaurin series for cos(x).
  • One participant emphasizes that they were specifically referring to the sigma notation part of the series.
  • Another participant argues that integrating the sigma notation is not feasible in the same way as integrating a function.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the integration of the sigma notation part of the Maclaurin series for sin(x). There are competing views on whether it is appropriate to integrate the series directly or if it should be treated differently.

Contextual Notes

Some assumptions about the nature of integration and the treatment of series in mathematical analysis are not fully articulated, leading to potential misunderstandings.

hawaiidude
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i was stumbled on a problem was i was looking over my book...i can do things like, differential 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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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?
 
integrate the sigma part
 
"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 "differential geometry, Fourier analsysis, advanced calculus", I don't see why you would think sin(x) was not integrable.
 
no was talking about the sigma notation part but o well ok
 
you can integrate the sigma notation no more than you can integrate the + symbol.
 

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