# 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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### Use Taylor Series to arrive at the expression for the forward approximation for a derivative [duplicate]

Use Taylor Series to arrive at the expression for the forward approximation for a derivative. $$f'(x)\approx \frac1h\left(-\frac32f(x)+2f(x+h)-\frac12f(x+2h)\right)$$ I'm not sure how to even go ...
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### Real Analysis: Bounds for derivatives using Taylor's Theorem

Suppose that $f''$ exists on [0,1] and that $f(0)=0=f(1)$. Suppose also that $|f''(x)|\leq K$ for $x\in(0,1)$. Prove that $|f'(1/2)|\leq K/4$ and that $|f'(x)|\leq k/2$ for $x\in(0,1)$. I'm trying to ...
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### Finding a Maclaurin series

I have a question here; suppose $f(x)= x^2\sin(x^3)$ By using the Maclaurin series for sine, find the Maclaurin series for $f$ I understand how to obtain the Maclaurin series for $f$ using the ...
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### Series expansion at infinity

I am trying to find to generalize the limit that involves all rational functions such as $\sum_{n=0}^{l}\frac{{a}_{n}{x}^{n}}{{b}_{n}{x}^{n}}$. I believe the best way of generalizing all of them is ...
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### Bounding the error for $e^{x+5y}$ taylor polynomial expansion

The exercise asks me to prove: $$|e^{x+5y}-P_1(x,y)|< \frac{3}{2}(x+5y)^2$$ when $x+5y<1$ I don't understand what's the exercise suggesting but I tried this: $e^{x+5y} - P_1(x,y)$ is just ...
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### Taylor series of a convolution

The derivation below is from Probability Theory: The Logic Of Science By E. T. Jaynes, chapter 7 "The Central Gaussian, Or Normal, Distribution", p.706 The Landon derivation. Text available online: ...
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### Taylor expansion of difference of functions

Is the taylor expansion of the difference of functions (more specifically the difference of the same function at different points) simply the difference of the taylor expansions? Since that may be ...
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### Estimating error using Taylor Polynomial

I have searched and read quite a bit on this subject but I can't get this last bit straight. Reading the other answers did not help me unfortunately for me. Anyway the problem: Suppose I have the ...
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### Is e^(-1/x) a flat function at x = 0?

Taylor series of e^(-1/x) at x = 0 shows that it is flat function on x= 0. But in every text on flat function I see the example of the function e^(-1/x^2) and not e^(-1/x). I am starting to think that ...
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### Prove that the series $\sum_{n=0}^\infty \frac{(-1)^n \cdot x^{2n}}{(2n)!}$ represents $\cos x$ for all values of $x$

guys. The question is as stated in the title: prove that the series $\sum_{n=0}^\infty \frac{(-1)^n \cdot x^{2n}}{(2n)!}$ represents $\cos x$ for all values of $x$ My doubt is quite theoretical:...
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### How large need n to be to ensure that Taylor polynomial around x=0 gives a value of sin(pi) which has an error of less than 0.001?

I've found different methods to calculate $n$, but all include that I test it for several $n$. Is it possible to make a general formula that gives me the answer without having to test it, or do I need ...
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I'm working on arithmetical analysis and more specifically on finite differences. I want to create a series consisting of the following terms : $$f(x_{0} + ph) = f_{0}+ p\Delta f_{0}+ \frac{p(p+1)}{2!... 1answer 53 views ### Taylor Series Expansion of  f(x) = \sqrt{x}  around  a = 4  guys. Here's the exercise: find a series representation for the function  f(x) = \sqrt{x}  around  a = 4  and find it's radius of convergence. My doubt is on the first part: I can't seem to find ... 1answer 19 views ### Write the indicated case of taylor's formula I have this problem: "Write the indicated case of Taylor's formula for the given function. What is the Lagrange remainder in each case? f(x) = \ln{x} a = 1, n = 6 " That's the information I ... 0answers 26 views ### Is there a way to separate this function? let f(\mathbf r_1,\mathbf r_2) = \frac{1}{|\mathbf r_1 - \mathbf r_2|^2 + a^2}. Is there a method to represent this as (a series of) separated functions in the form?: f = \sum \limits _i g_i(\... 0answers 35 views ### How to derive analog of power rule for other forms of the derivative? Introduction We'll be dealing with multiple forms of calculus here. So we'll use \operatorname{L_d}(f(x)) to refer to the additive derivative, \cfrac{df}{dx}, \operatorname{P_d}(f(x)) to refer ... 2answers 49 views ### What is the trick to Taylor expand this function to 4th order? The function is  u(x,y)= -x-y-xyu^3, and I want to Taylor expand u(x,y) around (0,0) in powers of x and y to 4th order. To first order, I differentiated implicitly, and the expansion is:$$u(x,y)...
Consider function f(x)=$\frac{1}{1-x}$, find the remainder term Rn(Z) of a function of x and n. I now know that $f^{(n)}(x)=\frac{n!}{(1-x)^{n+1}}$ and that $Rn(z)=\frac{f^{n+1}(z)}{(n+1)!}(x)^{n+1}$...
Let $f: {\mathbb R}^n \to {\mathbb R}^n$ be a continuously differentiable function. Then, we can rewrite its first-order Taylor expansion at $x \in {\mathbb R}^n$ for $h \in {\mathbb R}^n$ that \begin{...