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irinina [24]
3 years ago
7

Use the Taylor series you just found for sinc(x) to find the Taylor series for f(x) = (integral from 0 to x) of sinc(t)dt based

at 0. a.Give your answer using summation notation. b.Give the interval on which the series converges.
Mathematics
1 answer:
Marina CMI [18]3 years ago
7 0

In this question (brainly.com/question/12792658) I derived the Taylor series for \mathrm{sinc}\,x about x=0:

\mathrm{sinc}\,x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}

Then the Taylor series for

f(x)=\displaystyle\int_0^x\mathrm{sinc}\,t\,\mathrm dt

is obtained by integrating the series above:

f(x)=\displaystyle\int\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}\,\mathrm dx=C+\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}

We have f(0)=0, so C=0 and so

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

which converges by the ratio test if the following limit is less than 1:

\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{2n+3}}{(2n+3)^2(2n+2)!}}{\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}}\right|=|x^2|\lim_{n\to\infty}\frac{(2n+1)^2(2n)!}{(2n+3)^2(2n+2)!}

Like in the linked problem, the limit is 0 so the series for f(x) converges everywhere.

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5 0
3 years ago
What is m pls help i will mark brainly
-Dominant- [34]

Answer:

x=18

Step-by-step explanation:

Hey There!

So if you did not know all of the angles in a triangle will add up to equal 180

So we use the equation

180=2x+3x+5x

and solve for x

step 1 combine like terms

2x+3x=5x

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so now we have

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step 2 divide each side by 10

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hope this helps and if you have any more questions feel free to ask :)

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