# 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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### $(1+1/n)^n (1+x/n)$ decreasing iff $x\geq\frac{1}{2}$

$(1+1/n)^n (1+x/n)$ decreasing iff $x\geq\frac{1}{2}$. My question is how to prove $(1+1/n)^n (1+x/n)$ decreasing, starting from $n=1$. For large $n$, it is easy to show by Taylor expansion.
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### Why the expected value of error Taylor series approximation around the mean is zero?

I came across the following sentence in Paul Wilmott introduces quantitative finance. ... a random variable S (in our example the stock price), then ... using a Taylor series approximation around the ...
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### If Complex Numbers Describe a Circle and Split-complex Numbers Describe a Hyperbola, Can One Make a Hypercomplex Number System to Describe any Shape?

I was thinking about other complex-like systems the other day, and I decided to define a number $o$ such that $o^2 = 1, o \ne \pm 1$. I wondered if there was a formula like Euler's formula for this ...
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### How to prove cosine law using the power series expansion of cosine and sine?

How to prove the equation $\cos(x + y) = \cos(x) \cos(y) -\sin(x) \sin(y)$ using the power series expansions \begin{equation*} \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}, \qquad \sin(x)...
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### Find particular solution for $(D^2+1)y=e^{a \cos x}$, where a is an arbitary constant.

I tried solving the problem as in the image by taking partial integrals and also by series expansion, but it is becoming more complex to continue and find a close form of it. Please solve it.
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1 vote
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### Pointwise upper bound on $|f(x+y+z)-[f(y)+f(z)+f'(y)(x+z)]|$ where $f(x) = |x|^{p-1}x$

Let $f(x) = |x|^{p-1}x$ for some $2 \leq p \leq 3$. I've seen the pointwise estimate $$\left| f(x+y+z)-[f(y)+f(z)+f'(y)x + f'(y)z] \right| \lesssim |f(x)| + |f'(x)y| + |f'(z)x| + |f'(z)y|$$ in Lemma 3....
### Maclaurin series of $\frac{x^2}{1- x \cot x}$
I wonder if there is an explicit formula for the Maclaurin expansion of $\frac{x^2}{1 - x \cot x}$. We know an explicit formula for $1- x \cot x$. Due to the continued fraction formula for $\tan x$, ...