Question

If f (n)(0) = (n + 1)! for n = 0, 1, 2, , find the Maclaurin series for f.

Answer 1

$f(x)=\displaystyle\sum_{n=0}^\infty\frac{f^{(n)}(0)}{n!}x^n=\sum_{n=0}^\infty\frac{(n+1)!}{n!}x^n=\sum_{n=0}^\infty(n+1)x^n$

Note that, by the ratio test, this converges for

$\displaystyle\lim_{n\to\infty}\left|\frac{(n+2)x^{n+1}}{(n+1)x^n}\right|=|x|<1$

Integrating once, you have

$\displaystyle\int f(x)\,\mathrm dx=\int\sum_{n=0}^\infty(n+1)x^n\,\mathrm dx=\sum_{n=0}^\infty(n+1)\int x^n\,\mathrm dx$
$\displaystyle\int f(x)\,\mathrm dx=\sum_{n=0}^\infty\frac{(n+1)x^{n+1}}{n+1}+C=\sum_{n=0}^\infty x^{n+1}+C$

Given that $f(0)=0$, it's clear that $C=0$, and so

$\displaystyle\int f(x)\,\mathrm dx=\displaystyle\sum_{n=0}^\infty x^{n+1}=x\sum_{n=0}^\infty x^n=\frac x{1-x}$

Differentiating both sides recovers $f(x)$:

$f(x)=\dfrac{(1-x)(1)-x(-1)}{(1-x)^2}=\dfrac1{(1-x)^2}$

Answer 2

The Maclaurin series for f is found by using the ratio test, integration, and differentiation is given below.

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

What is a series?

A series is a sum of sequence terms. That is, it is a list of numbers with adding operations between them.

The Maclaurin series for f will be given below.

$f(x) = \sum_{n=0}^{\infty} \dfrac{f^{(n)} (0)}{n!} \ x^n = \sum_{n=0}^{\infty} \dfrac{(n+1)!}{n!} \ x^n = \sum_{n=0}^{\infty} (n+1) \ x^n$

By the ratio test, this converges for

$\displaystyle \lim_{n \to \infty} \left| \dfrac{(n+2)x^{n+1}}{(n+1)x^n} \right| = |x| < 1$

Integrating once, then we have

$\rm \int f(x)dx = \int \sum_{n=0}^{\infty} (n+1)x^n dx = \sum_{n=0}^{\infty} (n+1 ) \int x^ndx\\\\\\\int f(x)dx = \int \sum_{n=0}^{\infty} \dfrac{(n+1 ) x^{n+1}}{n+1} +C = \sum_{n=0}^{\infty} x^{n+1} +C$

Given that f(0) = 0, then the value of C will be 0. Then we have

$\rm \int f(x)dx = \sum_{n=0}^{\infty} x^{n+1} = x\sum_{n=0}^{\infty} x^n = \dfrac{x}{1-x}$

On differentiating, we have

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

More about the series link is given below.

https://brainly.com/question/10813422