# Tagged Questions

Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

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### Intuitive derivation of Taylor expansion?

I was looking up the derivations of Catalan numbers, and one derivation (probably the most famous) involves generating functions that leads to: $$C(x) = \frac{1}{2} (1 - \sqrt{1-4x})$$ And then this ...
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### When can you “plug in” a function $g(x)$ directly into a taylor expansion of a function $f(x)$ to get the expansion of $f(g(x))$, specifics below

I have asked a couple of questions Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives Computing the taylor series of $f(x)=(x+2)^{1/2}$ around $x=2$ up to ...
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### Is the mathematical syntax correct here (Taylor-polynomial)?

Say I'm supposed to create the $2^{nd}$ degree Taylor-polynomial of $f(x) = \cos x$ at $x_{0} = 0$ I'd like to know if the syntax is correct, how I solved this little task. We have defined the ...
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### A problem related to Taylor series [on hold]

Prove that there exists a constant $C \in \mathbb{R}$ so that $$\sum_{k=1}^{N}\frac{1}{k}=\log{N}+C+O\left(\frac{1}{N}\right)$$
For every differentiable function $f(x)$, is $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}$ always true and can be written? The only thing we have to be concerned is just whether ...