# Questions tagged [taylor-expansion]

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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### How to arrive at this sufficient condition for convergence of Taylor Series?

So I have been studying the convergence of Taylor series from Tom M Apostol. There is a theorem which states the sufficient condition for convergence of Taylor series. Quoting the theorem we've ...
0answers
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### Under what condition, a function can be upper bounded by its second-order Taylor expansion?

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$, and $g(x) = f(a) + \nabla f(a)^\top (x-a) + \frac{1}{2}(x-a)^\top \nabla^2f(a) (x-a)$ be its second-order expansion at $a$. We know that if $f$ is concave,...
1answer
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### Laurent series of a cosine function

I want to find Laurent series of the complex function $$f(z) = \cos\left(\frac{z^2-4z}{(z-2)^2}\right)$$ at $z_0=2$. I will be thankful for any hints.
3answers
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### Having problem finding an error that is less than what is given.

so I have a problem with a problem that has to do with Taylor polynomials. Here is what they want me to do: "Find a value do cos(1/10) with means of Taylor polynomial method, with an error less than ...
3answers
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### How to prove relation between $(1-x)^k$ and $(1-x^k)$?

I have been trying to prove or disprove following inequality: $(1-x)^k \geq (1-x^k)$ $\forall 0 \leq x \leq 1$ and $k \in \mathbb{N}$. I thought about taking $\log$ on both sides and comparing but ...
4answers
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### Taylor series for exp(exp(x)) using just the power series for exp(x) [duplicate]

I'm trying to figure out how to calculate the power series for exp(exp(x)) using exp(x) and then to write down the first few terms. I have the answer for the terms but I don't know how they arrived at ...
1answer
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1answer
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