Question

What is the Taylor series of $x{e^x}$ ?

Answer 1

Hint: A function's Taylor sequence is an infinite sum of terms expressed in terms of the derivatives of the function at a single point. Near this point, the function and the number of its Taylor series are identical for most common functions.

Complete step-by-step solution:
The Maclaurin series was named after Colin Maclaurin, who used this special case of Taylor series extensively in the $18th$ century. A Maclaurin series is a power series that can be used to approximate a function $f(x)$ for input values close to zero if the values of the successive derivatives of the function at zero are known.
Let us start solving the series by considering the Maclaurin series for ${e^x}$
We know that,
\[
  {e^x}\, = \,1\, + \,x\, + \,\dfrac{{{x^2}}}{2}\, + \,\dfrac{{{x^3}}}{3}\, + \,\dfrac{{{x^4}}}{4}\, + \,..... \\
  \,\,\,\,\,\, = \,\sum\limits_{n = 0}^\infty {\dfrac{{{x^n}}}{{n!}}} \\
 \]
So, if we multiply $x$,to the above equation we get:
\[ x{e^x}\, = \,x\left\{ {1\, + \,x\, + \,\dfrac{{{x^2}}}{{2!}}\, + \,\dfrac{{{x^3}}}{{3!}}\, + \,\dfrac{{{x^4}}}{{4!}}\, + \,...} \right\} \\
  \,\,\,\,\,\,\,\, = x\, + \,{x^2}\, + \,\dfrac{{{x^3}}}{{2!}}\, + \,\dfrac{{{x^4}}}{{4!}}\, + \,\dfrac{{{x^5}}}{{5!}}\, + \,...\,\,\,\, \\ \]
On further solving we get,
\[ \,\,x{e^x} = \,\sum\limits_{n = 0}^\infty {x.\dfrac{{{x^n}}}{{n!}}} \\
  \,\,\,\,\,\,\,\,\,\, = \,\,\sum\limits_{n = 0}^\infty {\dfrac{{{x^{n + 1}}}}{{n!}}} \\ \]
The Taylor series of $x{e^x}$is \[\,\,\,\sum\limits_{n = 0}^\infty {\dfrac{{{x^{n + 1}}}}{{n!}}} \]
Additional Information:
The $nth$ Taylor polynomial of the function is a polynomial of degree $n$ generated by the partial sum of the first $n + 1$ terms of a Taylor sequence. Taylor polynomials are function approximations which become generally better as n increases. Taylor's theorem calculates the error introduced by such approximations in terms of numbers.
If a function's Taylor series is convergent, the sum of its Taylor polynomials is the limit of the infinite sequence of Taylor polynomials. Even if a function's Taylor series is convergent, it can vary from the sum of its Taylor series.

Note: A Maclaurin series is the extension of a function's Taylor series around zero. A Maclaurin series can be used to approximate a function, find the antiderivative of a complicated function, or compute a sum that would otherwise be impossible to compute.