# Get Ahead in Taylor Series: Expert Guidance and Real-World Applications

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.

Calculus 2Answered question
drogaid1d8 2022-11-23

## Show Taylor series of $f\left({x}^{2}\right)$

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 $x=9$.

Calculus 2Answered question
Hallie Stanton 2022-11-18

## If the Taylor Series of $\mathrm{ln}\left(x\right)$ is known:$\mathrm{ln}\left(x\right)=\left(x-1\right)-\frac{1}{2}\left(x-1{\right)}^{2}+\frac{1}{3}\left(x-1{\right)}^{3}-\frac{1}{4}\left(x-1{\right)}^{4}+\frac{1}{5}\left(x-1{\right)}^{5}-...$Can one find the Taylor series of$f\left(x\right)=\frac{x}{1-{x}^{2}}$by manipulating the Taylor series of $\mathrm{ln}\left(x\right)$?

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:$f\left(z\right)={z}^{3}\mathrm{sin}3z$ around ${z}_{0}=0$$f\left(z\right)=\frac{z}{\left(1-z{\right)}^{2}}$ around ${z}_{0}=0$

Calculus 2Answered question
Davirnoilc 2022-11-16

## Prove that if $f$ is defined for $|x| and if there exists a constant $B$ such that$|{f}^{n}\left(x\right)|\le B$for all $|x| and $n\in \mathbb{N}$, then the Taylor series expansion :$\sum _{n=0}^{\mathrm{\infty }}\frac{{f}^{\left(n\right)}\left(0\right){x}^{n}}{n!}$converges to $f\left(x\right)$ for $|x|.

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${e}^{\left(-{t}^{2}\right)}=\sum _{n=0}^{\mathrm{\infty }}\frac{\left(-{t}^{2}{\right)}^{n}}{n!}$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 be the unique function that satisfies and for all . Find the Taylor series for about .Wouldn't the value of every derivative at just be ? So how does a Taylor series even exist?If , then can the Taylor series of be the same as that of ?

Calculus 2Answered question
Rosemary Chase 2022-11-03

## Let $f\left(x\right)$ and $g\left(x\right)$ be two Taylor series such that:$f\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\left(-1{\right)}^{n}a\left(n\right){x}^{n}$and$g\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}b\left(n\right){x}^{n}$for $a\left(n\right)>0$ and $b\left(n\right)>0$.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 $f\left(x\right)=\left(x-1{\right)}^{3}$ 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 $f\left(x\right)$ about a point $x=a$ equivalent to the the Maclaurin series of another function $h\left(x\right)=f\left(x+a\right)$?

Calculus 2Answered question
tikaj1x 2022-10-31

## A function $f$ is defined as$f\left(x\right)=\left\{\begin{array}{rlr}& \frac{cosx-1}{{x}^{2}}& for\phantom{\rule{thinmathspace}{0ex}}x\ne 0\\ & \frac{-1}{2}& for\phantom{\rule{thinmathspace}{0ex}}x=0\end{array}$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 ${P}_{2}\left(x\right)$ for the function $f\left(x\right)=\left(4x-11{\right)}^{3/2}$ 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=?

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

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!