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Galina-37 [17]
1 year ago
13

Find a power series representation for the function

Mathematics
2 answers:
lyudmila [28]1 year ago
6 0

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

Mars2501 [29]1 year ago
5 0

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.

<h3>Find the power series representation for the function:</h3>

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

#SPJ1

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