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# Worked example: coefficient in Taylor polynomial

Finding the coefficient of the term containing (x+2)⁴ in the Taylor polynomial centered at x=-2 of x⁶-x³.

## Want to join the conversation?

• Is there a video explaining how 0!=1!=1? Is that a definition?
• That's not a definition. Turns out that the factorial function has an extension to all real numbers(and even complex numbers) except negative integers, it's denominated Gamma function and evaluating Gamma of 0 you get 1.
You can understand that idea better with this video:
• where is the x+2 coming from? Is it just part of the question/ is it arbitrary? Could the question be asking what the coefficient is if containing (x+19)^4 and have it be the same answer?
• since it is says centered at x = -2 the value of a in (x-a) will be -2. hence, (x-(-2))=x+2
• Why would you ever make a series out of a function that is all ready a polynomial?
• Hello, I recently started following your tutorials because my university lecturer does not explain appropriately. However, so far you are much better but in some questions in our notes, they ask for the 42nd term (really huge terms that cannot be done manually), and as soon as I went through the lecture notes, it appeared like black magic. Could you please do a tutorial about finding extremely far values for the Taylor/Maclaurin series? Also, how do you know if the centre is 0, or x itself is 0 if it does not specify it in the question?
• It's good to note that some functions have repeating derivatives; for example, any derivative of e^x is simply e^x, or as another example, a derivative of sinx which is divisible by 4 is always going to be sinx (so, the 40th derivative of sinx would be sinx, and if the value were say 42 like in your case, the derivative could be represented as 40 +2, and since the 40th derivative is sinx, it is essentially just the second derivative of sinx, or -sinx.) I hope that helps; however, if the function is really messy, then you may be best off using something like Wolfram Alpha.