# 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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### How can I prove that $e^x \cdot e^{-x}=1$ using Taylor series?

When proving $e^x.e^{-x}=1$ by using Taylor series, there are infinite many terms of $e^x$ and $e^{-x}$. Is there any fancy way to combine terms by terms to show that eventually it is equal to $1$?
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### what's the taylor serie and it's convergence

I have this problem: What is the Taylor series of $\sqrt{x}$ at $x_0 = 4$. What is its interval of convergence? I am stuck and I can not finish it. Any idea on how to do that? Thank you
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First of all, why do we take the absolute value of both sides? What is the point/reason? and for (x_n - z)^2 , isnt it always positive? Second of all, do you call this a recurrence relation? for the ...
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### How many n derivatives do you take for Taylor series to be accurate? [closed]

How many derivatives must we take to consider some Taylor series an accurate reflection of a function?
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### Solution of the functional equation $g(x)g(z) = g(x+z)+g(x-z)$

What is the solution for the following functional equation? $g(x)g(z) = g(x+z)+g(x-z)$ The solution given is: $g(z) = 2\cos(z)$. In the derivation of the result (using Taylor expansion), there is ...
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### Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
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### Coefficients of power series

After expansion, we have $$(x_1+x_2+\dots+x_n)^m=a_1x_1^m+a_2x_1^{m-1}x_2+\dots$$ where $x_{()}$ is the variable and constant indices $n>m$. What is the expressions of all these possible ...
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### Why is the Taylor series of $1/\sqrt{1-4q^2}$ popping up in my recursively defined triangle of polynomials?

While answering this question I stumbled on some nice (inexplicable) observation where a recursively defined sequence of polynomials turned out to coincide with some Taylor development I'll start ...
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### How do you Taylor expand the log likelihood function of the Poisson distribution?

This question is an extension to this previous question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a ...
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### Prove the series expansion

Prove that $$(1+x)^\frac{1}{x}=e-\frac{e}{2}x+\frac{11e}{24}x^2-\frac{7e}{16}x^3....$$ where e is exponenial , can any one give a proof...I tried with series expansion i could not get it.
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### Prove the following using Maclaurin's theorem

Prove that $$\log(1+e^x)=\log 2+\frac{1}{2}x+\frac{1}{8}x^2-\frac{1}{192}x^4......$$ I have tried doing it. Tell me if you think the question is wrong
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### How to show $1 +x + x^2/2! + \dots+ x^{2n}/(2n)!$ is positive for $x\in\Bbb{R}$?

How to show $1 + x + \frac{x^2}{2!} + \dots+ \frac{x^{2n}}{(2n)!}$ is positive for $x\in\Bbb{R}$? I realize that it's a part of the Taylor Series expansion of $e^x$ but can't proceed with this ...
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### Taylor series of $\ln(x+2)$

I try to determine the Taylor series of $\ln(x+2)$. Since I know $\ln(1-x) = \sum_{n=1}^{\infty} \frac{x^n}{n}$, I suppose I can rewrite, \begin{align} \ln(x+2) &= \ln(1-(-(x+1)))=-\sum_{n=1}^{\...
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### Taylor series and radius of convergence: $\sqrt{x}$ with centre $x = 16$?

I've been struggling with this question for a while now and getting nowhere with it. Could someone please help me out? Assuming that the function has a power series expansion about the given point, ...
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### Can I integrate then differentiate this power series to derive the same result as the binomial series expansion?

I've tried something but I'm not getting the right answer, so I'm wondering why it doesn't work. I want to taylor expand $\frac1{z^2}$ about some point $a\in\mathbb{C}$. Here's what I did: \begin{...
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### What could the ratio of two sides of a triangle possibly have to do with exponential functions?

Name says it all. The two seem so unrelated? What's more, if I'm not mistaken the exponential version contains an imaginary part. I'm kind of ignorant about imaginary numbers, but does this mean that ...
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### How to perform taylor expansion with numerical differentiation formula

I am attempting to perform taylor expansion on the following numerical differentiation formula: $f'''(0) = \frac {−f(−3h/2) + 3f(−h/2) − 3f(h/2) + f(3h/2)) }{ h^3 }$ Over the reference interval [−3h/...
I am wondering what is the second order Taylor expansion of a vector-valued function $f(x):\mathbb{R}^M\rightarrow \mathbb{R}^N$. I know that the gradient of a vector-valued function is a Jacobian ...