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Galina-37 [17]
2 years ago
13

Find a power series representation for the function

Mathematics
2 answers:
lyudmila [28]2 years 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]2 years 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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PLSSS HELP
Dennis_Churaev [7]

The correct standard form of the equation of the parabola is:

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<h3 /><h3>What is a parabola?</h3>

An equation of a curve that has a point on it that is equally spaced from a fixed point and a fixed line is referred to as a parabola. The parabola's fixed line and fixed point are together referred to as the directrix and focus, respectively. It's also crucial to remember that the fixed point is not located on the fixed line. A parabola is a locus of any point that is equally distant from a given point (focus) and a certain line (directrix). An essential curve of the coordinate geometry's conic sections is the parabola.

For the given question,

Vertex of parabola is (-3,3)

Thus, the equation of the parabola is:

(x-h)^{2}=4(y-k)

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Learn more about parabolas here:

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