Suppose that we have the Taylor series centered at x=0 (Maclaurin series) defined below.?

👨🏻‍🏫 學術台數學

[匿名]

2015-04-05 11:37 pm

Suppose that we have the Taylor series centered at x=0 (Maclaurin series) defined below.

1-(x^2/2!)+(x^4/4!)-(x^6/6!)+...+(-1)^n*(x^2n/(2n)!)

what is f'''''(0) = ? (f^(5)(0))

what is f''''''''''(0) = ? f^(10)(0)

I have no ideas how to approach this.. please help...

更新1:

can anyone also answer my other question? https://ca.answers.yahoo.com/question/index?qid=20150405143448AAkB25i

回答 (3)

ted s

2015-04-05 11:45 pm

✔ 最佳答案

hopefully you could recognize that this is cos x....5th derivative is -sin x, 10th is - cos x

or you recognize that the nth derivative of x^n = n!

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[匿名]

2015-04-05 11:47 pm

sum_0^infinity (-1)^n*x^(2n)/(2n)! = cos(x)

But for your question it shall just be

sum_5^5 (-1)^n*x^(2n)/(2n)! = (-1)^5*(x^10)/(10!)

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[匿名]

2015-04-05 11:38 pm

Yes.

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