# Tagged Questions

54 views

### Sum of Taylor Series

I have the converging series: $$1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+...$$ and I'm trying to find its sum when x = .9. I know this is the Taylor series for some function$f(x)$, and that I can ...
33 views

### Series representation of function with fractions, logarithms, squares and cosines.

I'm looking for a series representation for $$\dfrac x{x^2+(\log \cos x)^2}$$ Where $x\in(0,\pi/2)$ Note: Both finite and infinite series are accepted. I have tried taylor series, but it requires ...
176 views

### How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?

In an older question here in MSE I've asked for the term for the "slicing" of a power series in partial series and have learned that it is "multisection". I' ve been looking at the behaviour of the ...
174 views

28 views

60 views

### Showing Taylor Series for $f(x) = e^{-x^2}$ converges to $f$

Show Taylor Series for $f(x) = e^{-x^2}$ converges to $f$ I am stuck because when taking the (n+1) th derivative of f, I do not see a general pattern. Meaning I am having difficulty in bounding ...
48 views

### Function whose power series coefficients contain logarithms

Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients $a_n$ are expressions containing $\log n$ or something similar?
626 views

### Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
35 views

### upper bound for the series $S (x) = \sum_{k=1}^{\infty} \frac{(x_n-n)^k}{k!}$ from $|x_n -(n+1)|\leq x$.

I've been trying to find a tight upper bound for the series $$S (x) = \sum_{k=1}^{\infty} \frac{(x_n-n)^k}{k!}$$ in terms of finite value $x\in \mathbb R$, where: 1- $\{x_n\}$ is a sequence of a ...
110 views

### Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that ...
12 views

### Taylor's expansion of the singular part of an analytic function

Assume $f$ is analytic on the annulus $R_1<|z-a|<R_2$. Assume $R_1<r<|z-a|$. Define $f_2$ by $$f_2(z)=\frac1{2\pi i}\int_{|x-a|=r}\frac{f(x)dx}{x-z}$$ $f_2$ is analytic on $|z-a|>r$. ...
20 views

### Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
25 views

### Two case about convergent series

Could you help me to prove analytically that ? I started to study Taylor Series and I'm lost. Thanks in advance.
190 views

33 views

### Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n}$$ And that this Taylor series has a radius of ...
78 views

### If $\displaystyle \sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n= 0$ for any $n$ [closed]

Suppose that $f(x)=\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}$ for all $x$ with the radius of convergence $R>0$. If $\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n=0$ for any $n$.
30 views

### Maclaurin series stuck at finding $L_n$

I need to develop Maclaurin serie of $f(x)=\frac{1}{(1-x)^2}$ I found all the derivative, and all the zero values for the derivatives. I come up with that : ...
66 views

### Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
31 views

### Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
40 views

66 views

### Proof of $(1-e^{ix})^{-1}$

In G.H. Hardy's book 'Divergent Series' there is a claim that $(1-e^{ix})^{-1} = \frac {1}{2} + \frac {1}{2} i \cot (\frac {1} {2} x)$ I, for the life of me, can't get past showing that ...
422 views

### Series expansion of square root function

Could I please know how to expand: $$\sqrt{4-3x^2}$$ I simplified it to $\sqrt{1-\frac34x^2}$ but the question is if I let $x = x^2$, what will the coefficient of, say $x^5$ or $x^{10}$ be in the ...
59 views

### Intuition behind Taylor/Maclaurin Series

** This is a different question than Intuition explanation of taylor expansion? ** I understand some of the intuition behind a Taylor/Maclaurin expansion. More specifically, I understand that adding ...
87 views

### Taylor expansion for a multivariable function

\begin{align} T(x_1,\dots,x_d) &= \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \cdots \sum_{n_d = 0}^\infty \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + ...
79 views

### High Order Derivative Using Maclaurin Series

Use the Maclaurin series to solve the following: $$\frac{d^6}{dx^6}(x^4e^{x^2})$$ I got about halfway through the problem before getting stuck. I am not sure how to solve it... Any advice? Also, ...
36 views

### Taylor Series Calc 2

I am not sure how to find a series representation for the natural log. If anyone can show me some helpful steps to solve this problem it would be greatly appreciated. What is the Maclaurin series ...
43 views

### Taylor Series of $\frac{1}{1-x^2}$ about x=2
I am trying to form a taylor series of the following: $\frac{1}{1-x^2}$ about $x=2$ I tried factoring the equation such that it becomes the following: $\frac{1}{{(1+x)}{(1-x)}}$ I tried to ...