Recent questions in Taylor Series

Calculus 2Answered question

Whothoromapyfb 2023-01-07

What is the taylor series of ${\mathrm{cos}}^{2}\left(x\right)$ ?

Calculus 2Answered question

chchchchinacjn 2022-12-18

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

If a function has a nth degree Taylor series approximation, we can use the Lagrange form of the remainder to calculate the maximum value of the error of approximation. If the series is also an alternating series, we can use the Alternating Series Estimation Theorem to get another maximum value of the error of approximation. It is not guaranteed that the two maximums will always be the same.

If a function has a nth degree Taylor series approximation, we can use the Lagrange form of the remainder to calculate the maximum value of the error of approximation. If the series is also an alternating series, we can use the Alternating Series Estimation Theorem to get another maximum value of the error of approximation. It is not guaranteed that the two maximums will always be the same.

Calculus 2Answered question

Barrett Osborn 2022-11-22

The Taylor Series of Gamma Function exists or not.

Calculus 2Answered question

MISA6zh 2022-11-18

Converging series:

$1-x+\frac{{x}^{2}}{2!}-\frac{{x}^{3}}{3!}+...$

find its sum when $x=9$.

$1-x+\frac{{x}^{2}}{2!}-\frac{{x}^{3}}{3!}+...$

find its sum when $x=9$.

Calculus 2Answered question

Hallie Stanton 2022-11-18

If the Taylor Series of $\mathrm{ln}(x)$ is known:

$\mathrm{ln}(x)=(x-1)-\frac{1}{2}(x-1{)}^{2}+\frac{1}{3}(x-1{)}^{3}-\frac{1}{4}(x-1{)}^{4}+\frac{1}{5}(x-1{)}^{5}-...$

Can one find the Taylor series of

$f(x)=\frac{x}{1-{x}^{2}}$

by manipulating the Taylor series of $\mathrm{ln}(x)$?

$\mathrm{ln}(x)=(x-1)-\frac{1}{2}(x-1{)}^{2}+\frac{1}{3}(x-1{)}^{3}-\frac{1}{4}(x-1{)}^{4}+\frac{1}{5}(x-1{)}^{5}-...$

Can one find the Taylor series of

$f(x)=\frac{x}{1-{x}^{2}}$

by manipulating the Taylor series of $\mathrm{ln}(x)$?

Calculus 2Answered question

kemecryncqe9 2022-11-17

Good way of memorizing Taylor series for common functions?

Calculus 2Answered question

spasiocuo43 2022-11-17

Find and state the convergence properties of the Taylor series for the following:

$f(z)={z}^{3}\mathrm{sin}3z$ around ${z}_{0}=0$

$f(z)=\frac{z}{(1-z{)}^{2}}$ around ${z}_{0}=0$

$f(z)={z}^{3}\mathrm{sin}3z$ around ${z}_{0}=0$

$f(z)=\frac{z}{(1-z{)}^{2}}$ around ${z}_{0}=0$

Calculus 2Answered question

Davirnoilc 2022-11-16

Prove that if $f$ is defined for $|x|<r$ and if there exists a constant $B$ such that

$|{f}^{n}(x)|\le B$

for all $|x|<r$ and $n\in \mathbb{N}$, then the Taylor series expansion :

$\sum _{n=0}^{\mathrm{\infty}}\frac{{f}^{(n)}(0){x}^{n}}{n!}$

converges to $f(x)$ for $|x|<r$.

$|{f}^{n}(x)|\le B$

for all $|x|<r$ and $n\in \mathbb{N}$, then the Taylor series expansion :

$\sum _{n=0}^{\mathrm{\infty}}\frac{{f}^{(n)}(0){x}^{n}}{n!}$

converges to $f(x)$ for $|x|<r$.

Calculus 2Answered question

Tiffany Page 2022-11-14

In higher dimensions, is the derivative (jacobians,gradients etc.) defined using taylor series or taylor series formula proved through derivatives ?

Calculus 2Answered question

Ty Moore 2022-11-12

Integrate the Taylor series

${e}^{(-{t}^{2})}=\sum _{n=0}^{\mathrm{\infty}}\frac{(-{t}^{2}{)}^{n}}{n!}$

term-by-term to obtain the Taylor series for erf (error function) about $a=0$.

${e}^{(-{t}^{2})}=\sum _{n=0}^{\mathrm{\infty}}\frac{(-{t}^{2}{)}^{n}}{n!}$

term-by-term to obtain the Taylor series for erf (error function) about $a=0$.

Calculus 2Answered question

Tiffany Page 2022-11-12

Are there instances when a Taylor series and a Laurent series of the same function about the same point ever equal?

Calculus 2Answered question

Aliyah Thompson 2022-11-11

Let $\text{}F(x)\text{}$ be the unique function that satisfies $\text{}F(0)=0\text{}$ and $\text{}F\prime (x)={\displaystyle \frac{\mathrm{sin}({x}^{3})}{x}}\text{}$ for all $\text{}x$. Find the Taylor series for $\text{}F(x)\text{}$ about $\text{}x=0$.

Wouldn't the value of every derivative at $\text{}0\text{}$ just be $\text{}0\text{}$ ? So how does a Taylor series even exist?

If $\text{}F(0)=0$, then can the Taylor series of $\text{}F(x)\text{}$ be the same as that of $\text{}{F}^{\prime}(x)\text{}$ ?

Wouldn't the value of every derivative at $\text{}0\text{}$ just be $\text{}0\text{}$ ? So how does a Taylor series even exist?

If $\text{}F(0)=0$, then can the Taylor series of $\text{}F(x)\text{}$ be the same as that of $\text{}{F}^{\prime}(x)\text{}$ ?

Calculus 2Answered question

Rosemary Chase 2022-11-03

Let $f(x)$ and $g(x)$ be two Taylor series such that:

$$f(x)=\sum _{n=0}^{\mathrm{\infty}}(-1{)}^{n}a(n){x}^{n}$$

and

$$g(x)=\sum _{n=0}^{\mathrm{\infty}}b(n){x}^{n}$$

for $a(n)>0$ and $b(n)>0$.

Can we extract the asymptotic behavior of these two taylor series for $x\to \mathrm{\infty}$?

$$f(x)=\sum _{n=0}^{\mathrm{\infty}}(-1{)}^{n}a(n){x}^{n}$$

and

$$g(x)=\sum _{n=0}^{\mathrm{\infty}}b(n){x}^{n}$$

for $a(n)>0$ and $b(n)>0$.

Can we extract the asymptotic behavior of these two taylor series for $x\to \mathrm{\infty}$?

Calculus 2Answered question

Kareem Mejia 2022-11-02

What is the Taylor Series for $f(x)=(x-1{)}^{3}$ centered at $x=0$? What is the radius of convergence?

Calculus 2Answered question

Mark Rosales 2022-11-02

Is it always valid to say that the Taylor series of some function $f(x)$ about a point $x=a$ equivalent to the the Maclaurin series of another function $h(x)=f(x+a)$?

Calculus 2Answered question

tikaj1x 2022-10-31

A function $f$ is defined as

$$f(x)=\{\begin{array}{rlr}& \frac{cosx-1}{{x}^{2}}& for\phantom{\rule{thinmathspace}{0ex}}x\ne 0\\ & \frac{-1}{2}& for\phantom{\rule{thinmathspace}{0ex}}x=0\end{array}$$

Using the first three non zero terms of the Taylor series for cosx about $cos\phantom{\rule{thinmathspace}{0ex}}x$, write the first three non zero terms of the Taylor series for $f$ about $cos\phantom{\rule{thinmathspace}{0ex}}x$.

$$f(x)=\{\begin{array}{rlr}& \frac{cosx-1}{{x}^{2}}& for\phantom{\rule{thinmathspace}{0ex}}x\ne 0\\ & \frac{-1}{2}& for\phantom{\rule{thinmathspace}{0ex}}x=0\end{array}$$

Using the first three non zero terms of the Taylor series for cosx about $cos\phantom{\rule{thinmathspace}{0ex}}x$, write the first three non zero terms of the Taylor series for $f$ about $cos\phantom{\rule{thinmathspace}{0ex}}x$.

Calculus 2Answered question

Rubi Garner 2022-10-28

Determine the second-degree Taylor polynomial $${P}_{2}(x)$$ for the function $$f(x)=(4x-11{)}^{3/2}$$ expanded about x=5

Calculus 2Answered question

Aryan Lowery 2022-09-29

The Taylor series for f(x)=x^3 at -4 is \sum_{n=0}^{\infty} c_n(x+4)^n. Find the first few coefficients.

c_0=?

c_1=?

c_2=?

c_3=?

c_4=?

c_0=?

c_1=?

c_2=?

c_3=?

c_4=?

Calculus 2Answered question

Greyson Landry 2022-07-30

Find the Taylor polynomial of degree n=4 for each function expanded about the given value of ${x}_{0}$.

$f(x)={x}^{5}+4{x}^{2}+3x+1,{x}_{0}=0$

$f(x)={x}^{5}+4{x}^{2}+3x+1,{x}_{0}=0$

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Taylor Series are an essential tool in mathematics that allow the expansion of functions into an infinite sum of terms. It is a powerful way to approximate complicated functions, or to find a formula for the nth derivative of a function. The Taylor Series formula is based on the concept of derivatives, and is made up of an infinite number of terms that represent the successive derivatives of the function at a particular point. Working with Taylor Series can be tricky, so practicing Taylor Series problems is a great way to gain a better understanding of the formula and its applications. Fortunately, we have plenty of equations and practice problems to help students learn Taylor Series. So if you are struggling with Taylor Series, don't hesitate to look for help!