Let f(x)=\tan^{-1} x, x \in \mathbb{R}, then the sequence \{f^{(n)}(0)\}

2alr8w

2alr8w

Answered question

2022-01-22

Let f(x)=tan1x,xR, then the sequence {f(n)(0)} is unbounded. True or false?

Answer & Explanation

Karly Logan

Karly Logan

Beginner2022-01-23Added 11 answers

True is the answer, and it gets short if the Taylor series
f(x)=arctanx=k=0(1)k2k+1x2k+1
is available right from scratch. Then f(n)(0) can be read of from the Taylor coefficients: The non-zero values satisfy
f(2k+1)(0)(2k+1)!=(1)k2k+1
thus one has
f(n)(0)={0  is even  (1)n12(n1)!nn is odd.

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