# 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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### Proofs for Taylors theorem and other forms

Let $f \in C^k[a,b]$.Show that for $x,x_0 \in [a,b]$, $$f(x)=\sum\limits_{j=0}^\mathbb{k-1}{{1\over j!}f^{(j)}(x_0)(x-x_0)^j}+{1\over k!}{\int_{x_0}^x f^{(k)}(t)(x-t)^k \,dt}$$ and after this use this ...
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### Proof that $\oint_r d(x,N + n) < 0$?

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a ...
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### how to find taylor serie for 1/z with |z| > 0?

I have the following and I need to give the Laurent development for |z| > 0. The Laurent development in this form : and to give few a(n) coefficients How can it be done? normally we use the ...
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### Taylor series, identify radius of convergence

I have the following function : I need to find it's radius of convergence with z0 = 0. The function is analytic everywhere except where 1 + sin(iz) = 0 (to my ...
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### Laurent-series expansion of $\frac{1}{(e^z-1)^2}$ about $z=0$

I am studying for exams in complex analysis and taking a look at past papers. This comes up often or an integral of the given function along a certain curve, which is actually the same thing since the ...
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### Find the Taylor series about $x = 1$ for $f(x) = \dfrac{1}{(x − 2)^2}$ . [closed]

Find the Taylor series about $x = 1$ for $f(x) = \dfrac{1}{(x − 2)^2}$ . Express your answer in sigma notation, simplified as much as possible. This is a practice question that I am having trouble ...
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### sin(x+y^2) taylor expansion little oh error term degree >3

I am trying to understand example 3.4.5 in John and Barbara Hubbard's second edition of Vector Calculus, Linear Algebra, and Differential Forms. It provides the taylor expansion of $sin(x+y^2)$ by ...
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### Taylor series expansion?

How to find the Taylor series expansion of $$(1+x)^{1/x}$$ I tried with the Taylor series but unable to solve it. Help me out. Hints or anything that sort will be helpful.
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### Does Cauchy's estimate imply analyticity?

Komatsu says here (Proc. Japan Acad. Volume 36, Number 3 (1960), 90-93) that a smooth function which satisfies Cauchy's estimate is analytic. How does one prove this? Surely, if Cauchy's estimates ...
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### Taylor expansion for $\arcsin^2{x}$

I stumbled upon this particular expansion that was included in this post. $$\displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$ This caught ...
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### An infinite sum in the product of sines

This is an undergrad or lower level question I need help with. Evaluate $$\quad \sum_{n=1}^{\infty} \sin{\left(\frac{a}{3^n}\right)}\sin{\left(\frac{2a}{3^n}\right)}$$ where a is just some real ...
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### Is this Taylor series correct taken correctly? Confused reasoning

I have $dx/dy=-ay, x(0)=1$ initial value problem. Then $x(y)=\frac{x(0)}{0!}y^0+ \frac{x'(0)}{1!}y+\frac{x''(0)}{2!}y^2=1+(-a)y+a^2y^2...$
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### Intuition behind power series

I keep seeing power series throughout mathematics disguised in all different shapes, yet I can't seem to put my finger on what is really fundamentally being expressed here. Some examples: Arabic ...
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### Find Maclaurin expansion of $y=2^x$ to $x^4$

Find Maclaurin expansion of $$y=2^x\text{ to } x^4$$ This is my try. We have $\displaystyle 2^x=e^{x\ln 2} =\left[1+\frac{x^2}2+\frac{x^3}6+\frac{x^4}{24}+o(x^4)\right]^{\ln 2}$ with $o(x^4)$ is ...
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### Derivative of a definite improper integral

-The derivative with respect to beta, for the following definite integral is required. g = $\int_\beta^{\sqrt(\beta^2 +1}$ $erfc(\gamma z)/\sqrt(z^2 - \beta^2)$dz -I am using the leibniz formula ...
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### Solving initial value problem with Taylor Series expansion $dx/dt=x^2, x(0)=1$

I have series of homework questions that call for using Taylor series to solve initial value problems so can someone solve this example and explain what they are doing? I also want to know what ...
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### What is the nth derivative of $\dfrac{1}{\sqrt{1 + x^2}}$

I'm trying to find a general formula for the $n$th derivative of $$\dfrac{1}{\sqrt{1 + x^2}}$$ I got up to, \begin{eqnarray*} g^{(0)}(x) &=& g(x) \\ g^{(1)}(x) &=& \dfrac{1}{(1 + ...
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### Proving $f=0$ if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ . [duplicate]

Let $f\in C^{\infty}[-1,1]$ and let $M$ be a constant such that $|f^{(j)}(x)|\le M$ $\forall j\in \Bbb{Z}_{+}$ and $x\in [-1,1]$. Prove that if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ then $f=0$. I ...
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### Taylor series expansion and radius of convergence

The problem is: Expand the given function using Taylor's expansion around $a=1$. $f(x)=(5x-4)^{-7/3}$, and then find the radius of convergence of the obtained series. Hint : Write the nth derivative ...
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### Estimate probability of event using moments of a distribution or a Taylor expansion involving the moments

Let's say we have four moments $(\mu_1, \mu_2, \mu_3, \mu_4)$ of a probabilty distribution of a random variable $X$ and the goal is to get the probability $\rm{P}(X \leq t)$ for a certain value of $t$....
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### Limit Question involving logarithmic taylor expansion

I need to evaluate the limit for part of my proof: $$\lim_{n \to \infty}\left(1-\dfrac{1}{\eta^{x}} \right)^n$$ My attempt: \begin{align*} \lim_{n \to \infty} F_{\eta_n}(x) &= \lim_{n \to \...
### How to show $\sum_0^\infty \frac{x\lambda^x} {x!} = \lambda e^\lambda$?
I know that $\sum_0^\infty \frac{\lambda^x} {x!} = e^\lambda$, but I'm having a really difficult time dealing with the extra $x$.