0
votes
0answers
16 views

Find the minimum number of terms in the series it would take before the error would be guaranteed to be less than 10^-9

Consider the function $f(t)=\ln t$ about the point $t_0=1$. Find the minimum number of terms in the series it would take before the error would be guaranteed to be less than $10^{-9}$.
1
vote
0answers
24 views

Will this method find the taylor expansion of ANY function $f(x)$?

Polynomials are themselves Taylor expansions, correct? ex. $4x+5x^2+3 = 3+4x+5x^2 +0x^3 +0x^4 + \dots$ I'm assuming has no closed form besides $\sum_{n=0}^{2}(3+n)x^n + \sum_{n=3}^{\infty}0x^n$ but ...
0
votes
0answers
12 views

The error function in the taylor's theorem for taylor series

I was reading taylor's theorem at wikipedia and at some point they say that $f(x)$ can be written as a function related to its linear approximation $P_1 = f(a) + f'(a)(x-a)$. This is a very simple ...
1
vote
0answers
22 views

If I have an infinite series, how do I know that the digits I calculated are rigth?

For example, there are infinite series for $\pi$, $e$, $\phi$... But if I sum a finite ammount of terms, I get an approximation for the series. How do I know how much correct digits of this ...
1
vote
5answers
198 views

Why are Maclaurin series useful if we can only use them for such a small range of numbers?

Okay, I am beginning to get how Maclaurin series work, but what I don't understand is why they are useful. Why would you want an infinite expansion for a series that works for such few values (only ...
0
votes
2answers
33 views

Cannot expand $\sin(2x^2-4x+3)$ at $x_0 = 1$

Trying to expand $\sin(2x^2 - 4x+3)$ at $x_0 = 1$ to the $O(x-x_0)^n$. After substitution $t = x - 1 $, the problem becames $$\sin(2t^2+1) \text{ at } t_0 = 0$$ While we know that $$\sin(s) = ...
1
vote
2answers
24 views

Importance of the first term in a Taylor series

Suppose you have a function $f(x)$ whose Taylor series can be represented as the power series $$a_0 + a_1x^2+a_2x^4+...$$ If you are told that for $x\in\mathbb{R}_+$, $$a_0 + a_1x^2 + a_2x^4 + ...
2
votes
0answers
154 views
+200

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
1
vote
1answer
24 views

Finding a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$.

I'm trying to find a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$. So I am using the Maclaurin Series representation of $f(x)=\ln(1+x)$ which is ...
3
votes
5answers
96 views

When $a\to \infty$, $\sqrt{a^2+4}$ behaves as $a+\frac{2}{a}$?

What does it mean that $\,\,f(a)=\sqrt{a^2+4}\,\,$ behaves as $\,a+\dfrac{2}{a},\,$ as $a\to \infty$? How can this be justified? Thanks.
0
votes
1answer
11 views

Use the power series representations of functions to find the taylor series of $\frac{1}{5+x'}$ at center = -6.

I am trying to find the taylor series of $f(x)=$ $\frac{1}{5+x'}$. And I cannot seem to get how to find the taylor series using the method I've been using for other functions. Another thing that's ...
0
votes
1answer
24 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
1
vote
2answers
25 views

Sum Representation of log(1 + x)

$\log(1+x) = \sum_{k=1}^{\infty} \left(\dfrac{x}{1+x}\right)^{k} \dfrac{1}{k} = \sum_{k=1}^{\infty} \left(1 - \dfrac{1}{1+x}\right)^k \dfrac{1}{k}$ Why is this true? The most sum representation of ...
1
vote
1answer
61 views

Taylor Polynomial of $f(x)=\cos(x)\cdot\sin(x)$

How would I calculate the third maclaurin/taylor polynomial on $\cos(a) \cdot \sin(b)$, Do I use the product rule when I calculate the derivatives? I don't know where to start or read about it, been ...
1
vote
1answer
36 views

The $n$th-derivative of $q(x) = x^4 - 8x^3 - 4x^2 + 3x - 2$, where $n \le 4$

Some factors will be $\frac{4!}{(4-n)!}\cdot a_nx^n, \frac{3}{(3-n)!} \cdot a_{n-1}x^{n-1}, \ldots, \frac{1!}{(1-n)!} \cdot a_0x^0$, but the lowest degree one will always become zero in the next ...
0
votes
1answer
41 views

Help finding n-order Maclaurin polynomial

EDIT AND PLEASE NOTE: I DON'T want solutions that are nicer or more elegant but presume knowledge of other infinite series and/or don't come from the nth-derivative because I'm precisely studying how ...
0
votes
1answer
39 views

The accuracy of approximating $ f(x) = x^{2/5}$ for $0.9 \le x \le 1.1$ using the cubic Taylor polynomial

For the equation $ f(x) = x^{2/5}$, $a=1$, $n=3$, $0.9 \le x \le 1.1$ I was able to approximate f by the following Taylor polynomial: $$ F_3(x) = 1 + \frac2 5 (x-1) - \frac3{25}(x-1)^2 + ...
0
votes
0answers
24 views

Finding an optimal path for minimizing an integral.

Let $x,y$ be real numbers. Let the function $f(x,y)$ be real-entire in both $x$ and $y$. Thus $f(x,y)$ is a real-entire Taylor series in the variables $x,y$. How the find a non-intersecting path ...
0
votes
1answer
13 views

Maclaurin series accuracy

Find an $n_1$ such that the $n_1$th-order Taylor polynomial for $\sin(x)$ about $x=0$ gives an approximation of $\sin(x)$ with an error of less than $5\cdot 10^{-10}$, for all $x$ between $0$ and ...
0
votes
0answers
18 views

Taylor expansion of a scalar function

I have an expression on the form $$ \sum_{i=1}^N{\rho_i}f(\mathbf x+\mathbf c_i)\mathbf c_i $$ where $\rho_i$ is a scalar, $f(\mathbf x+\mathbf c_i)$ a scalar function of the vector quantities ...
3
votes
1answer
54 views

manipulations with Taylor expansions for log and sinh

How could we derive the equality $$ \frac14 \sum_{m=1}^\infty \frac1m \frac{1}{\sinh^2 \frac{m\alpha}2} = - \sum_{n=1}^\infty n\log (1-q^n)$$ where $q=e^{-\alpha}$ ?
6
votes
0answers
77 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider ...
1
vote
2answers
26 views

Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
0
votes
2answers
42 views

Why cant we do substitution in differentiation but is ok in taylor series?

I have the same question 10 year ago when i was studying high school. I dont understand it and i give up the math. 10 year ago, i need to work with calculus during work and this question come to find ...
0
votes
1answer
27 views

Prove error bound using Taylor's series Error term (Bound doesn't seem to make sense)

I have to prove that at least seven terms must be used in the Taylor series estimation of x - sin(x) in order for the error to be <= $10^{-9}$. This doesn't seem correct however. This series is ...
0
votes
1answer
30 views

Infinite series expansion of $\arcsin (x)$ and $\arccos (x)$

How to find the infinite series expansion of $\arcsin (x)$ and $\arccos (x)$?
3
votes
4answers
48 views

Taylor polynomial of $\frac{1}{2-x}$

Can someone show how to find the Taylor polynomial of $\frac{1}{2-x}$? I tried this: $\frac{1}{2-x}=\frac{1}{1-(x-1)}$ and then use that $ \ T_n(\frac{1}{1-x})=1+x+\dots +x^n.$ But this gives ...
2
votes
4answers
158 views

How can I accurately compute $\sqrt{x + 2} −\sqrt{x}$ when $x$ is large?

How can the values of the function $f(x) = \sqrt{x + 2} −\sqrt{x}$ be computed accurately when $x$ is large? I have tried using Matllab. I am not able to understand when $x$ will be large.
1
vote
1answer
54 views

Differentiability of the remainder in Taylor's theorem

Suppose we have a function that's differentiable $m$ times over $[a,b]$, we have $a< \alpha < x < b$ and $n < m$. Then $$ f(x) = \sum_{i = 0}^{n-1} \frac{f^{(i)}(\alpha)}{i!}(x - ...
6
votes
4answers
186 views

Closed-forms of infinite series with factorial in the denominator

How to evaluate the closed-forms of series \begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, ...
0
votes
1answer
31 views

Maclaurin series for $\frac{1}{|1+x|}$

I believe that there is no Maclaurin Series for $\frac{1}{|1+x|}$ as the latter is not differentiable at $x=-1$. However, would it be appropriate for me to refer $\frac{1}{|1+x|}$ as 'not a smooth' ...
2
votes
1answer
46 views

$f$ differentiable and $f(0)=f(1)=0$. , prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$

Let $f$ be differentiable on $[0,1]$ and $f(0)=f(1)=0$. Also, we know $|f''(x)| \le A$ on $(0,1)$, prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$ I'm guessing I should use taylor ...
2
votes
1answer
35 views

Evaluating $\ln(\cos x))$ using Taylor expansion

Evaluate $\ln(\cos x)$ at $x_0=0$ and with the order of $n=4$. Noticing that $\ln(\cos x) = \ln(1+ \cos x - 1)$ we can use $\ln(1+x)$ Taylor series. Now, I've read I should use: $$\ln(1+x) = x - ...
1
vote
1answer
78 views

What is the limit regarding $a$

What is the limit of : $$ \lim_{x\to 0} \frac{\sin(ax) - \ln(1-2x)}{e^{ax}-1-2x-2x^{2}}$$ I did this with Maclaurin, because my exam is about solving these with MacLaurin. Gave $$\lim_{x\to 0} ...
1
vote
1answer
32 views

Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
0
votes
2answers
29 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
2
votes
1answer
38 views

taylor series expansion, derivatives not continuous

As a part of an excercise I am supposed to find the Taylor series expansion for $(1-t)^{\frac{1}{2}}$ on $[0,1]$. According to the remainder theorem: ...
0
votes
2answers
65 views

Taylor series $(x+1)^{\frac{1}{3}}$

Complete the Maclaurin polynomial of degree three for $(x+1)^{1/3}$. I have completed the first two derivatives of this function and thus have coefficients but am not certain how to put them into ...
3
votes
2answers
95 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
2
votes
2answers
37 views

Evaluating a limit with Taylor Series

I would like to find the following limit using Taylor Series: $$\lim_{x\to0}\frac{6\sinh x-6x-x^3}{x^4(6+x^2)\sinh x}.$$ Now my question is the following: How do I know exactly how to approximate ...
0
votes
0answers
24 views

Lagrange's form of the remainder vs Cauchy's form

So far (while practicing exercises) I've used Lagrange's form of the remainder. Is there a situation when Cauchy's form comes in handy while Lagrange's form fails for some reason? Is there a rule of ...
1
vote
0answers
52 views

Given a function $f$, find the largest $n$ such that $f(x)/x^n$ can be defined at $x=0$ to become differentiable there

Let $f(x) = \ln\left(\frac{x^2}{2}+1\right)+\cos x -1$. Find the largest $n\in\Bbb{N}$ such that there is $C\in\Bbb{R}$ such that: $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C ...
1
vote
1answer
38 views

Using Maclaurin approximation to find the limit of $(\ln(1+x^2)-\ln(1-x^2))/(e^{x^2}-e^{-x^2})$ as $x\to 0$

I have this assignment: $$\lim_{x\to0}\frac{\ln(1+x^2)-\ln(1-x^2)}{e^{x^2}-e^{-x^2}}$$ And by using the Maclaurin approximations I get this: ...
3
votes
3answers
61 views

Show $\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$

I'm reading a proof which aim to show that: $$\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$$ the Taylor expansion of $\ln(1+x)$ is (not by chance): $$x - ...
2
votes
1answer
72 views

Evaluate a limit using Taylor series

Let $$\lim\limits_{x\to 0}\frac{({\ln(1+x) -x +\frac{x^2}{2})^4}}{(\cos(x)-1+\frac{x^2}{2})^3}$$ Now, I know that I should utilize Taylor polynomial. $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...
1
vote
2answers
41 views

Multplication of series

My textbook is taking about the Cauchy product and I don't quite understand it and it says that when multiplying series, the sum of the third one is equal to the product of the sums of first two ...
1
vote
1answer
70 views

Taylor series and Lagrange's remainder f(x)=$e^x$

In my textbook the Lagrange's remainder which is associated with the Taylor's formula is defined as: $R_{n}(x)= \frac{(x-a)^n}{n!} f^{(n)}(a + \vartheta (x-a))$, for some $\vartheta$ $\in$ <0 ,1> ...
0
votes
1answer
71 views

Calculating Laurent Series of Complex Function

How does one alternate the Bernoulli number series expansion $$\frac x{e^x - 1}=\sum_{n=0}^{\infty}\frac{B_nx^n}{n!}$$ To calculate the Laurent Series centered at 0 in the annulus of convergence of ...
0
votes
2answers
41 views

Two-dimensional Taylor linearisation

I have to perform a first order taylor expansion of a function $f(\vec{x}) = f(x+u,y+1)$ at the point $\vec{a} =(x,y)$. My solution reads $$ f(\vec{x}) \approx f(x,y) + \left( \begin{matrix} ...
0
votes
1answer
79 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 ...