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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23 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, ...
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2answers
54 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 $\frac{x^3}{e^x-1}\approx\frac{x^3}{1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\dots}$ $~~~~~~~~= ...
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2answers
69 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 ...
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2answers
48 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 ...
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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
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170 views
+100

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 ...
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0answers
22 views

a function with many branch points : the radius of convergence of its Taylor series

How can I be convinced that if a (locally holomorphic) function $f(z)$ has many branch points, say at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$, and all of the weirdest type, then the radius of ...
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1answer
21 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 ...
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1answer
25 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: $$f^{(n)}(z)=(-1)^n n!/z^{n+1} \Rightarrow f^{(n)}(-1)=-n!$$ ...
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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 $k \in \mathbb{N}$: ...
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1answer
34 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) $e^{e^z}$ ...
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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
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3answers
41 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
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1answer
33 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
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1answer
281 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 ...
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0answers
17 views

Find T8(sin(x+x^4) and f(7)(o)

How do I approach this problem? what I did was find the taylor polynomial of sinx of order 8 and then replaced everything with (x+x^4). In order to find the 7th derivative do I have to expand the ...
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0answers
29 views

Show that $f(x)=\frac{1}{x^2}$ is real-analytic in $(0,∞)$

Show that:$$f(x)=\frac{1}{x^2}$$ is real-analytic in $(0,∞)$. I'm having trouble using Taylor's theorem to prove this
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0answers
28 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 ...
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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 ...
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2answers
66 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 ...
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1answer
23 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 $, ...
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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: ...
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0answers
16 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
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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
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1answer
18 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 ...
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2answers
31 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 ...
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2answers
119 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
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2answers
45 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
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1answer
17 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 ...
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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
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1answer
26 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 $z$?
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46 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.
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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?
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2answers
43 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
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0answers
57 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 ...
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1answer
17 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 ...
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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 ...
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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 ...
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2answers
1k views

How local is the information of a derivative?

I have read it a thousand times: "you only need local information to compute derivatives." To be more precise: when you take a derivative, in say point $a$, what you are essentially doing is taking a ...
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2answers
47 views

Marsden's definition of Taylor Series

How does the following definition of Taylor polynomials: $f(x_0 + h)= f(x_0) + f'(x_0)\cdot h + \frac{f''(x)}{2!}h^2+ ... +\frac{f^(k)(x_0)}{k!}\cdot h^k+R_k(x_0,h),$ where ...
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1answer
38 views

Determine the first four non-zero terms in the power series expansion about $x=0$ for the general solution: $\left(2x-3\right)y''-xy'+y=0$

$$\left(2x-3\right)y''-xy'+y=0$$ First I found the first to derivatives of the following power series: $$y(x)=\sum_{n=0}^{\infty}a_nx^n$$ $$y'(x)=\sum_{n=1}^{\infty}na_nx^{n-1}$$ ...
2
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1answer
26 views

Taylor Series, Approximation of a Function

a) Find the first 5 terms of the Taylor series for $f(x)=1/\sqrt{x}$ centered at $a=4$ b) use the result from a) to estimate $1/\sqrt{3}$ and compare it to calculated value My attempt at the ...
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1answer
41 views

Compute the Taylor Series for $f\left(x\right)=\ln\left(1+x^2\right)$ about $x= 0$

I'm very confused by this question. Can you provide me with hints as to how to get started with this one? $f\left(x\right)=\ln\left(1+x^2\right)$ about $x= 0$ Do I just use the Taylor Series ...
2
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1answer
24 views

about truncated taylor expansions

I have a question about expanding $e^u$into a truncated taylor series where $u$ is itself a truncated Taylor series (in my example $u$ is expansion of $-\frac{\log(1+t)}{t}$, up to term $O(t^3)$), it ...
4
votes
2answers
203 views

Applying Taylor theorem on a linear map

I found the following in a stack of practice problems but had trouble dealing with it: Consider a linear map $A:C^\infty(\mathbb{R}^n)\rightarrow \mathbb{R}$ such that: If $f\in ...
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1answer
78 views

Truncating a taylor expansion for a recurrence relation?

Let's say I have a function $N$ whose future value at a time $t + t_{d}$ obeys the relation $N(t + t_{d}) = A(t)N(t)$ where $A(t)$ is also a function of $t$ whose value can be calculated. One can ...
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1answer
18 views

Taylor expanding to leading order

I've had a lot of trouble finding a reduced form of the solutions here to the leading order: $$\omega_{1,2}=-\frac{1}{2}(1+k+\epsilon) \pm \frac 12 \sqrt{(1+k+\epsilon)^2-4k\epsilon}$$ The textbook ...
3
votes
1answer
67 views

Finding the value of $1.1^{82}$ using $(1+x)^{82}$ to a certain accuracy

I found this question in a book. How many terms of the Maclaurin expansion of $(1+x)^{82}$ are needed to guarantee finding a value of $1.1^{82}$ to an accuracy of $10^{-6}$? This is how I tried to ...
1
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1answer
28 views

Converting this summation into an integral

This summation includes a sum of n derivatives of the function f(x) at the point (c+d) / 2 I'm trying to convert a Taylor ...
2
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2answers
47 views

Determine the first three non-zero terms in the Taylor polynomial approximation for the initial value problem: $y''+\sin(y)=0$

Having trouble understanding how to solve this problem. Did I at least set it up correctly? $y''+\sin(y)=0,\;y(0)=1,\;y'(0)=0$ So assuming $y(x)=\sum_{n=0}^{\infty}a_nx^n$ then ...