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)

3
votes
2answers
34 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
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) $e^{e^z}$ ...
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
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?
2
votes
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 ...
24
votes
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 ...
4
votes
2answers
201 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 ...
2
votes
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 $z$?
0
votes
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 ...
0
votes
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
0
votes
0answers
27 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
1answer
27 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
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 ...
0
votes
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 $, ...
1
vote
0answers
110 views

Taylor's series question on bounds

$f(x) = f(y) + f'(y)(x-y) + \frac{f"(y) (x-y)^2}{2} + ....$ is the taylor's series. I am aware of bounds which specify the error in approximating $f$ with a polynomial. I want sufficient conditions ...
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
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) = ...
4
votes
2answers
118 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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
2answers
29 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 ...
0
votes
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 ...
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) = ...
1
vote
2answers
38 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 ...
2
votes
2answers
64 views

Taylor's theorem with remainder of fractional order?

Let $k\geq 1$. Consider Taylor's theorem. We know the Peano form and the mean-value form of the remainder term: Peano form of the remainder Let $f\colon (-\varepsilon,\varepsilon)\to\mathbb R$ be ...
2
votes
0answers
56 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 ...
1
vote
1answer
457 views

Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?

Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an ...
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?
0
votes
2answers
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.
0
votes
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 ...
0
votes
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 ...
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 ...
2
votes
2answers
46 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 ...
0
votes
1answer
36 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
votes
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 ...
-1
votes
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 ...
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
1answer
594 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
2
votes
2answers
42 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 ...
1
vote
1answer
22 views

propagation of error from product of Taylor Series

Say I have two functions $f(x)$ and $g(x)$, both of which I will be approximating with Taylor series $T_f(x)$ and $T_g(x)$ respectively. Lets say $f(x)$ is order $O(x^{n_1})$ and $T_f(x)$ has error of ...
0
votes
0answers
35 views

Derive the formula and its error from the basic rule $B(f: c,d)$

I understand Taylor series, for example I know the Taylor expansion for $f(a+bh)$ where $h \approx 0$ would be $$f(a+bh) = f(a) + bhf'(a) + b^2 h^2 f''(a)/2 + \cdots$$ But I have a problem that has ...
0
votes
1answer
25 views

Equivalence of Taylor series and its corresponding function and Axiom for infinite summation

Given a function $f(x)$ with a taylor series expansion, is it valid to say that $$f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(a)(x-a)^n$$ for all values of x irrespective of whether the taylor series ...
0
votes
3answers
26 views

show equality - binomial formula, taylor?

I am trying to show that this is true using the binomial formula or some taylor expansion: $\frac{1}{(1+\epsilon \sum\limits_{n=0}^\infty Z_n(t) \epsilon^n)^2} = 1 - 2Z_0\epsilon + ...
0
votes
1answer
22 views

Numerical analysis: what is the error term for the rule…?

The question goes: derive the error term for the rule $phi$ to approximate the third derivative of f(a). I have attached a screenshot I understand how to take the Taylor series in the hint, but the ...
1
vote
2answers
67 views

Estimating $\int_0^1f$ for an unknown Lipschitz $f$ to within 0.0001

A friend of mine has a Lipschitz function $f\colon [0,1]\to\mathbb R$ satisfying (Some more characters, and yet a few more...) $$|f(a)-f(b)| \le 5 |a-b| \qquad\text{for all }a,b\in[0,1],$$ ...
12
votes
4answers
3k views

On what interval does a Taylor series approximate (or equal?) its function?

Suppose I have a function $f$ that is infinitely differentiable on some interval $I$. When I construct a Taylor series $P$ for it, using some point $a$ in $I$, does $f(x) = P(x)$ for all $x$ in $I$? ...
4
votes
1answer
38 views

Maclaurin Series nth Derivative

Find $f^{(2016)}(0)$ if $f(x)=\sin(x^2)$. From the Maclaurin series, $$\sin(x^2)=\sum_{n=0}^\infty\frac{(-1)^nx^{4n+2}}{(2n+1)!}$$ Comparing coefficient, ...