Question

Find a power series representation for the function

Answer 1

Recall the power series expansions of $\sin(x)$ and $e^x$.

$\displaystyle \sin(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2n+1}$

$\displaystyle e^x = \sum_{n=0}^\infty \frac{1}{n!} x^n$

By substituting $3x$ for $x$ in the latter series, we have

$\displaystyle e^{3x} = \sum_{n=0}^\infty \frac{1}{n!} (3x)^n = \sum_{n=0}^\infty \frac{3^n}{n!} x^n$

Then the series expansion of $f(x)$ is

$\displaystyle f(x) = x^3 \sin(x) + e^{3x+2} \\\\ ~~~~~~~~ = x^3 \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2n+1} + e^2 \sum_{n=0}^\infty \frac{3^n}{n!} x^n \\\\ ~~~~~~~~ = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2(n+2)} + e^2 \sum_{n=0}^\infty \frac{3^n}{n!} x^n$

Answer 2

$f(x)=\sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n+1)!}x^{2(2n+2)} + e^{2} \sum^{\infty }_{n=0} \frac{1}{n!}3^{n}x^{n}$ is the power series representation for the function f(x) = x³sin(x) + e³ˣ⁺². This can be obtained by using power series representation of each terms, sin x, eˣ and substituting in the function.

Find the power series representation for the function:

In the question the given function is,

⇒ f(x) = x³sin(x) + e³ˣ⁺²

 

We know that series representation of sin x and eˣ are:

• $sin x = \sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n+1)!}x^{2n+1}$

• $e^{x} = \sum^{\infty }_{n=0} \frac{1}{n!}x^{n}$

   ⇒ $e^{3x} = \sum^{\infty }_{n=0} \frac{1}{n!}x^{n}$

             = $\sum^{\infty }_{n=0} \frac{1}{n!}3^{n}x^{n}$

Substituting the series representation in the function we get,

⇒ f(x) = x³sin(x) + e³ˣ⁺²

⇒ $f(x)=x^{3}\sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n+1)!}x^{2n+1} + e^{2} \sum^{\infty }_{n=0} \frac{1}{n!}3^{n}x^{n}$

$f(x)=\sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n+1)!}x^{2(2n+2)} + e^{2} \sum^{\infty }_{n=0} \frac{1}{n!}3^{n}x^{n}$

Hence $f(x)=\sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n+1)!}x^{2(2n+2)} + e^{2} \sum^{\infty }_{n=0} \frac{1}{n!}3^{n}x^{n}$  is the power series representation for the function f(x) = x³sin(x) + e³ˣ⁺².

Learn more about power series representation here:

brainly.com/question/11606956

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