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.

learn more… | top users | synonyms (2)

0
votes
2answers
27 views

When does Taylor series for g agree with g

For $g(x)=e^{-1/x^2}$ for x not equal to 0 and $g(0)=0$. How to show that the Taylor series for g about 0 agrees with g only at $x=0$? I know that the maclaurin series for g(x) is ...
5
votes
2answers
88 views

Characterizations of $e^x$

I've been thinking about the following problem for a while: (AFAIK) the 'exponential function', $e^x$ can be characterized as the unique solution to the following differential equation with initial ...
5
votes
5answers
109 views

Maclaurin Expansion for $e^{e^{z}}$ at $z=0$

I need to find terms up to degree $5$ of $e^{e^{z}}$ at $z=0$. I tried letting $\omega = e^{z} \approx 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$, and then substituting these first few terms ...
1
vote
0answers
42 views

Taylor series for $\arctan x$

We use $\frac{1}{1+x^2} = \sum_{n=0}^\infty (-1)^nx^{2n}$, where $|x|<1$ and integration yields $\arctan x = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{2n+1}$. And by the ratio test this series ...
1
vote
2answers
34 views

Check: Radius of Convergence of the Sum of these Complex Taylor Series

I just found the following Taylor series expansions around $z=0$ for the following functions: $\displaystyle \frac{1}{z^{2}-5z+6} = \frac{1}{(z-2)(z-3)} = \frac{-1}{(z-2)} + \frac{1}{(z-3)} = ...
1
vote
1answer
36 views

Find Maclaurin series for integral of $e^{z^2}$

I need to find a Taylor Series expansion of $\displaystyle \int_{0}^{z}e^{\zeta^{2}}d\zeta$ around $z=0$, which shouldn't be hard enough. Except that I can only integrate term-by-term if the Taylor ...
5
votes
2answers
80 views

Examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$

How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?
1
vote
3answers
36 views

Prove that the $c_{n}$ in $\frac{1}{1-z-z^{2}}=\displaystyle\sum_{n=0}^{\infty}c_{n}z^{z}$ satisfy a Fibonacci-like recurrence relation

I need to prove that the coefficients $c_{n}$ of the expansion $\frac{1}{1-z-z^{2}}\sum_{n=0}^{\infty}c_{n}z^{n}$ satisfy the recurrence relation $c_{0}=c_{1}=1$, $c_{n}=c_{n-1}+c_{n-2}$, by ...
0
votes
2answers
26 views

Taylor series for $\frac{1}{az+b}$ centered at $z=0$ by substitution

I need to find the Taylor series centered at $z=0$ (i.e., the Maclaurin series) for $\displaystyle \frac{1}{az+b}$, where $a,b \in \mathbb{C}$ and $b \neq 0$. I thought it would be good to start out ...
0
votes
1answer
42 views

Finding $f^{(12)}(0)$ with $f(x)=\log(e^{x^4}-2x^8)$

Here's how I proceeded: We have $f(x)=x^4+\log\left(1-2x^8e^{-x^4}\right),$ hence for all $x$ such that $-1\le2x^8e^{-x^4}<1$ the following holds: \begin{align} ...
0
votes
2answers
54 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
0
votes
1answer
31 views

Taylor expansion of $\sin(x-y)$

A question asks me to find the partial derivatives of $f: \mathbb{R}^2 \to \mathbb{R}$ with $f(x, y) = \sin(x-y)$ then asks me to give the taylor expansion of $f(\pi/2+h, k)$ in powers of $h$ and $k$ ...
1
vote
1answer
38 views

Find the first two terms in the perturbation expansion of the solution

I want to find the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ terms in the pedestrian expansion $y = y_0 + \epsilon y_1 + \epsilon ^2 y_2 + \dots$, where $y$ satisfies the following second order ...
0
votes
0answers
9 views

Expand or approximate entropy of a two-term Gaussian Mixture

Is it possible to create some expansion to approximate this $h(a)$ for $a>0$ near $a\rightarrow0$? $$N(x,v)\equiv\frac{1}{\sqrt{2\pi v}}e^{-\frac{x^{2}}{2v}}$$ $$ ...
1
vote
4answers
69 views

What is the general term for $e^x/(1-x)$

What id the taylor series expansion for $\frac{e^x}{1-x}$? I know that the series expansion for $e^x$ is the sum of $\frac{x^n}{n!}$ from $0$ to $infty$. But how can I account for the $1- x$ in the ...
0
votes
1answer
27 views

How do I work out the validity for a Maclaurin (power) series?

I cannot find the answer to this anywhere so I have decided to make a question. Given a Maclaurin series for a function, how can I quickly work out what the validity is for it? For example, ...
0
votes
2answers
56 views

Maclaurin series of $x^3/(e^x-1)$

how would i taylor expand $f(x)=\frac{x^3}{e^x-1}$ around $x=0$? I was thinking of writing $\frac{x^3}{e^x-1}\approx\frac{x^3}{1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\dots}$ $~~~~~~~~= ...
0
votes
2answers
71 views

Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding ...
1
vote
2answers
50 views

Why does the taylor series of $\frac {1}{\ln x}$ have a non-infinite radius of convergence?

Shouldn't the taylor series of a function be equal to that function for any input value? Why does this not work for the taylor series of $\frac {1}{\ln x}$ when $|x| \gt 1$? Edit: I do mean the ...
1
vote
3answers
34 views

Taylor's series and ln

Can someone explain to me how to find the $\lim \limits_{x \to 3} \frac{\ln|4-x|}{x-3}$ using taylor's series. Can someone explain the proof of $\ln|4-x|$ to power series please
7
votes
0answers
244 views

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
0
votes
1answer
24 views

Radius of Convergence of Taylor series without finding the series

How do you find the radius of convergence of a Taylor series for a function centered at point $z_0$ without actually finding the Taylor series? I know that we can use comparison test, ratio test or ...
0
votes
1answer
26 views

Expanding $1/z$ about $z=-1$ using Taylor series vs Power Series

I need to expand $1/z$ about $z_0=-1$. I decided to do it using both methods, which don't agree. Using Taylor: Finding coefficients: $$f^{(n)}(z)=(-1)^n n!/z^{n+1} \Rightarrow f^{(n)}(-1)=-n!$$ ...
3
votes
2answers
35 views

Exponential Taylor series with $k$ step

It is well-known that $$\sum_{n=0}^\infty \frac{x^n}{n!} = e ^x$$ or $$\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!} = \cosh x $$ My question is what we know about the sum for arbitrary $k \in \mathbb{N}$: ...
1
vote
1answer
39 views

Complex Taylor Series by substitution

I need to find the first few terms or so of the Taylor series centered at $z_0 = 0$ regarding these functions: a) $e^{z\sin z}$ b)$(1+z)^z = e^{z \ln (1+z)}$ c)$\cos (1 + z^3) $ d) $e^{e^z}$ ...
0
votes
0answers
15 views

Value of Taylor Approximation

I'm studying Trust Region Methods and there's one simple thing I'm finding it hard to wrap my head around. The premise of the method is to approximate $f(x)$ as a quadratic model: ...
2
votes
3answers
42 views

$n$-th Term for Maclaurin Series

On a Calculus BC test I had this morning, I had to find the first five terms and the $n$-th term of the following function: $$ f(x) = x \cos(3x)$$ According to my instructor, I could've manipulated ...
2
votes
1answer
34 views

How do I expand this function around zero?

The function is $$ \sqrt{\frac{\sin(x)}{x}} $$ I need to expand it to the order $x^2$ around $0$. The solution is supposed to be: $$ 1-\frac{x^2}{12}+\mathcal{O}(x^4) $$ How do I proceed?
12
votes
1answer
286 views

Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for ...
0
votes
0answers
31 views

Estimate sqrt(1.1) using Lagrangian formula and Taylor polynomials with error within 1/10^6

So I set f(x)=sqrt(1+x) and then went on to estimate the error for x=0.1 according to the Lagrangian formula will be f(n+1)(ξ)*0.1^(n+1)/(n+1!). I know 0<ξ<0.1 but I still cannot think of how ...
0
votes
0answers
43 views

Taylorseries of $\cos(x)e^x$

Lets consider $f:\mathbb R\rightarrow \mathbb R, f(x)=\cos(x)e^x$. I want to calculate the taylor-series around $x_0=0$ and I want to check if the taylor-series is equal to $f(x)$. The first ...
1
vote
2answers
68 views

Find a function for the infinite sum $\sum_{n=0}^\infty \frac{n}{n+1}x^n$

I need to find a function $f(x)$ which is equal to the sum $$ \sum_{n=0}^\infty \frac{n}{n+1}x^n, $$ for every $x\in \mathbb{R}$ for which the series converge. Now, using WolframAlpha, I've found the ...
0
votes
1answer
25 views

Finding the radius and the interval of convergence.

I usually use Ratio Test to find the radius and the interval of convergence. However, for this series, the ratio test does not work. If I use the ratio test, my answer is $|-2x+3|<1 $, ...
1
vote
0answers
18 views

Taylor Polynomial please explain order meaning? (example included)

When I am asked to find a Taylor polynomial of order 6th for example, does that mean that my answer HAS to include only powers of x up to 6? I am not sure how to solve the following example. Ex: ...
0
votes
0answers
17 views

Confirmation on a function satisfying specific conditions(Power Series)

I had a question, find a function that satisfies the following conditions and I have to use Power series. F is the function. 1) Domain is all reals, 2) $F''(x) = cos(x^2)$, 3) $F'(0) = 3$, 4) $F(0) = ...
0
votes
1answer
28 views

Taylor expansion, problematic integrand

Consider $$f(z) = \int_0^z \frac{1-\cos\sqrt{t}}{t}\mbox{d}t $$ Find its Taylor series at $a=0$. I was thinking about looking at the integrand, from which we would have: $$\frac{1-\cos\sqrt{t}}{t} = ...
0
votes
1answer
20 views

Upper bound on Taylor's series expansion of the exponential [closed]

I want a function $a:\mathbb{R}\to\mathbb{R}$ such that $$e^{x}\leq 1 + x + a(\epsilon) \frac{x^2}{2}\mbox{ for } |x|\leq \epsilon.$$ Is there a good choice such that $a(\epsilon)\to 1$ as ...
1
vote
2answers
32 views

How can we use series representation in limits?

How can we use series representation in limits? 1) We can write $\sin x$ as $$\sin x=\sum\limits_{i=0}^\infty \frac{(-1)^ix^{2i+1}}{(2i+1)!}.$$ How can we write this? For any given $\epsilon ...
4
votes
2answers
120 views

Infinite Sum without using $\sin\pi$

What's a purely algebraic way to prove that $\pi-\frac{\pi^3}{3!}+\frac{\pi^5}{5!}-\dots=0$? I'm sure that the first step is to write $\pi=4-\frac43+\frac45-\dots$, but I haven't been bold enough to ...
2
votes
2answers
46 views

Maclaurin serie of $\int_0^x\frac{sin(t)}{t}$

If $f(x)=\int_0^x\frac{\sin(t)}{t}$. Show that $$f(x)=x-\frac{x^3}{3*3!}+\frac{x^5}{5*5!}-\frac{x^7}{7*7!}+...$$ Calculate f(1) to three decimal places. Would you mind showing how to build this ...
0
votes
1answer
18 views

Determine radius of convergence of Taylor series of $f(z)$ at point $a$

Consider $$f(z) = \frac{z+e^z}{(z-1+i)(z^2-2)(z-3i)}, a=0 $$ As we can see it's quite ugly so I won't even try and develop a Taylor series of it at point $a=0$. I have noticed there are Four ...
1
vote
2answers
54 views

Sums of the series $1 + (x^2) / 3! +( x^4) / 5! +\cdots$

How can I compute sum of the series ; $$1 + \frac{x^2}{3!}+\frac{x^4}{5!}+\frac{x^6}{7!}+\frac{x^8}{9!}+\cdots$$ I tried to divide it to two pieces such that $$f(x) = ...
2
votes
1answer
27 views

Taylor vs Laurent series - cosines and sines

In general, why do we say that the Taylor series of sines and cosines are also Laurent series despite of the power of $z$?
0
votes
2answers
48 views

Maclaurin serie of $\frac{1}{(1-x)(1-2x)}$

Help me finding the Maclaurin serie of $$f(x) = \frac{1}{(1-x)(1-2x)} $$ in the easiest way (if there is one which you do not have to calculate a lot of derivatives) possible, please.
1
vote
2answers
53 views

Maclaurin series of $e^x\sin x$

Would you mind showing me a faster way of building Maclaurin series of $$f(x)=e^x\sin x$$ so I do not need to calculate a lot of derivatives?
1
vote
2answers
46 views

Reindexing Exponential Generating Function

I have an exponential generating function, and I need to double check what the teacher said, because I'm having trouble coming to the same result. Also, I need to verify what I am coming up with, and ...
3
votes
0answers
58 views

A sufficient and necessary condition of Taylor series

Let $f(x)$ be a $C^{\infty}$ function on $(-R,R)$. Prove that $f(x)$ can be expanded as its Taylor series at the point $x=0$ over the interval $(-R,R)$ if and only if there exists a positive function ...
0
votes
1answer
18 views

How toexpress $V=\frac{kq}{x-a}-\frac{kq}{x+a}$ in terms of $k,q,x,u$ in Taylor Series for the following condition?

The question calls $u=\frac{a}{x}$ and $u$ is the variable. So for Taylor Series, we express it in $f(x)=\sum^{\infty}_{k=0}\frac{f^k(0)}{k!}x^k$ However, one hint says all we need is geometric ...
1
vote
0answers
36 views

How to find function $F$ such that $F''(x)=\cos{x^2}$, $F'(0)=3$ and $F(0)=4$?

Here we want $F\in \Bbb{R}$. We use Taylor Series. I get $F''(x)=\cos{x^2}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k+1)!}(x^2)^{2k}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k+1)!}x^{4k}$ Integrating, we have ...
0
votes
0answers
28 views

Taylor Polynomial Approximtions

Answer Provided. Explanation needed. Hi, I am asked to construct a Taylor polynomial approximation that is accurate to within $10^{-3}$ over the indicated interval using $x_0=0$ with the following ...