Taylor series finding sin(x^2)+cos(x) from sin(x^2) and cos(x) alone

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Homework Help Overview

The discussion revolves around finding the Taylor series expansion for the expression sin(x^2) + cos(x) at x = 0, using the known series for sin(x^2) and cos(x). Participants are exploring the addition of these series and addressing discrepancies in the resulting coefficients.

Discussion Character

  • Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to combine the Taylor series for sin(x^2) and cos(x) and are questioning the accuracy of the resulting series coefficients, particularly for the x^6 term. There is a focus on the need for a common denominator when adding fractions.

Discussion Status

The discussion is active, with participants clarifying their understanding of polynomial addition and the Taylor series. Some guidance has been offered regarding the addition of terms and the importance of common denominators, though there is still uncertainty among participants about specific steps in the process.

Contextual Notes

Participants are navigating the complexities of infinite series and polynomial addition, with some expressing a need for more foundational understanding. There is an acknowledgment of potential confusion regarding signs and coefficients in the series.

IntegrateMe
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If I want to find the taylor series at x = 0 for sin(x^2)+cos(x)...

sin(x^2) = x^2 - x^6/3! + x^10/5! - x^14/7! ...
cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! ...

So why does sin(x^2) + cos(x) = 1 + x^2/2! + x^4/4! + 121x^6/6! ...?

Thanks!
 
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I get -121x^6/6!, other then that just give the fractions a common denominator and add.
 
yes, sorry, it is a minus sign...can you explain that more elaborately? I think I'm starting to understand, but I'm not quite there yet.
 
Well the 1 + x^2/2! + x^4/4! is hopefully obvious. As for the -121x^6/6!, you need to add those terms in your two taylor series which have an x^6 in them, so that is - x^6/3! and - x^6/6!. But before you can add these fractions you need to give them a common denominator. Remember 6! = 6*5*4*3*2*1 and 3! = 3*2*1.
 
Well the 1 + x^2/2! + x^4/4! is hopefully obvious.

Nope :\
 
Ok forgetting these series are infinite for a minute just consider the numbers you wrote without the ... . Then you are basically adding polynomials correct? So what do you do when you add polynomials well you add the x^2 terms together and you add the x terms together etc. Maybe you could be more specific about what you aren't understanding since I'm not sure how weak/strong your foundation is.
 

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