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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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A. 8/17<br> B. 15/17<br> C. 15/8<br> D. 8/15
Vlada [557]

Answer:

A)  8/17

Step-by-step explanation:

cos = adj/hyp

In this problem the unknown angle has a hypotenuse pf 17, 15 as opposite, and 8 as adjacent

8 0
2 years ago
The height of a tree trunk is 20 meters and the base diameter is 0.5 meter.
Temka [501]
Question a:

Mass = Density × Volume 
Density = Mass/Volume

Volume of the tree trunk (the shape of Cylinder) = Area of circular base × height
Volume = [πr²] × h
Volume = [π × 0.25²] × 20
Volume = 3.93 m³

Density = 380 kg/m³

Mass = Density × Volume
Mass = 380 × 3.93
Mass = 1493.4 kg

------------------------------------------------------------------------------------------------------------

Question b)

The growth ring = 4 millimeters = 4÷1000 = 0.004
New diameter = 0.5 + 0.004 = 0.5004
New height = 20 + 0.2 = 20.2

New volume = [πr²] × h
New volume = [π × 0.2502²] × 20.2
New volume = 3.97 m³



5 0
3 years ago
What is the arc measure of BDC in degrees?<br> (4k + 159)<br> P<br> (2k + 153)
andre [41]
<h2>Explanation:</h2><h2></h2>

The diagram is missing but I'll assume that the arc BDC is:

(4k + 159)^{\circ}

And another arc, let's call it FGH. measures:

(2k + 153)^{\circ}

If those arc are equal, then this equation is true:

(4k + 159)^{\circ}=(2k + 153)^{\circ} \\ \\ (4k + 159)=(2k + 153) \\ \\ \\ Solving \ for \ k: \\ \\ 4k-2k=153-159 \\ \\ 2k=-6 \\ \\ k=-\frac{6}{2} \\ \\ k=-3

Substituting k into the first equation:

\angle BDC=(4(-3)+159)^{\circ} \\ \\ \angle BDC=(-12+159)^{\circ} \\ \\ \boxed{\angle BDC=147^{\circ}}

3 0
3 years ago
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In creating the shooter-style game comparable to Space Invaders, we decided that each time the laser hits an object, the player
olya-2409 [2.1K]

Answer:

Outcome

Step-by-step explanation:

I'm not exactly certain but I believe the answer is outcome. For example: Each time the laser hits an object, the player gets a point. Because you're giving the cause, the laser hitting the object, and the effect, the player getting a point;its safe to assume that the element is outcome.

3 0
2 years ago
Consider the diagram.<br><br><br> Which line segment has the same measure as ST?
valentinak56 [21]
SR because any point on a perpendicular bisecting line is equidistant from the ends of the bisected segment.
8 0
3 years ago
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