# 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.

8,567 questions
Filter by
Sorted by
Tagged with
1 vote
59 views

### I do not know why there is a zero in this formula.

I am currently studying the chapter on Numerical Differentiation in my book. However, I have encountered a section where a zero appears in the expansion, and I'm unsure of the reason behind its ...
1 vote
22 views

### Approximating norm using Taylor series

I have a point $A =(r\cos\theta \sin\phi, r\sin\theta \sin\phi, \cos\phi)$ and $B =(nd_x, 0 ,md_z)$. My aim is to compute the Euclidean norm $||A-B||$. However, I am interested in an approximation ...
16 views

### Taylor Expanding a higher power of a function

I have a (sufficiently smooth) function $f$ that is volume-preserving (Jacobian has determinant $1$) and invertible. Given two integers $\ell, k \in \{0, 1, \ldots, T\}$ with $\ell > k$ it seems ...
48 views

### Integrating $\int_0^1 x^xdx$ [duplicate]

How would you calculate the following integral? $$\int_0^1 x^xdx$$ I write my answer below. Don't be shy to speak out if you find the mistakes (if there are any) or have a better solution.
39 views

177 views

### $f(z) = \sum_{n=0}^{\infty} f(n)^2 z^n$?

Im looking for functions $f(z)$ such that $f(z) = \sum_{n=0}^{\infty} f(n)^2 z^n = f(0)^2 + f(1)^2 z + f(2)^2 z^2 + f(3)^2 z^3 + ...$ and $f(n)$ are all real. And I wonder how fast $f(n)$ grows. I had ...
84 views

### Approximating $\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}$ [duplicate]

Consider the series $$\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}$$ How many terms of this series do I need to consider to have an approximation of $\pi$ accurate up to $10$ decimal places (for ...
1 vote
100 views

97 views

44 views

Basically, I approached this problem using the basic Taylor Series Expansion formula: $$f(z)=f(z_0) + \frac{f'(z_0)}{1!}(z-z_0)+\frac{f''(z_0)}{2!}(z-z_0)^2+\cdots$$ From there, I got stuck with $$f(z)... 3 votes 1 answer 142 views ### Taylor series of same function being different? I'm trying to expand ln(3+4x) into a taylor series \ln(3+4x) = \ln(1+2+4x) Let t = 2(1+2x) \ln(1+t) \approx t-\frac{t^2}{2} + \frac{t^3}{3} - \frac{t^4}{4} + \frac{t^5}{5} So then \ln(1+t) \... 0 votes 0 answers 45 views ### Point in use of Taylor Series to approximate functions in an age with computers? I hope this doesn't sound too vague or like I'm dismissing the use of Taylor Series entirely, I'm just curious about any proper real-world applications. Many times Taylor Series are shown-off as a ... 1 vote 2 answers 53 views ### In what sense, Taylor series at x=0 is better than Taylor series at x=a, and vice-versa? In the book `Calculus Early Transcendentals (6th Edition)' by James Stewart, on p. 739, the author writes: We have two series representations for e^x as follows:$$e^x=\sum_{n=0}^\infty x^n/n! , \...
1 vote
44 views

Background: Regular gradient descent can be written something like $x_{t + 1} = x_t - \eta g_t$, where $g_t$ is the gradient of the function we're trying to optimize. Problem: If we have a (symmetric, ...
572 views

62 views

50 views

### For all complex $|z| \neq 1$ : $\sum_{n=0}^{\infty} w_n \frac{z^n}{1+z^n+z^{2n}+z^{3n}+z^{4n}} = \sum_{n=0}^{\infty} u_n \frac{z^n}{1+z^n+z^{2n}}$?
Ok I am a bit confused. So here comes a question, Consider a maclaurin series for $f(z)$ $$f(z) = \sum_{n=0}^{\infty} f_n z^n$$ where $f(z)$ has a radius of exactly $1$. $f(z)$ may or may not have a ...
I noticed a pattern between matrix and Taylor series of a finite continued fraction function. However, I don't know how to prove it or why they are related. Let $$f_{1}(z)=\frac{1}{-z-1}$$  f_{2}(...