Consider the sequence described by xn 11n n 0 1 2 a find the...

Consider the sequence described by xn = 1/(1+n), n = 0, 1, 2, ... a) Find the suitable function g(x) so that the sequence can be generated by means of repeated substitution in the form xn+1 = g(xn), n = 0, 1, 2, ... b) Determine the rate of convergence of the sequence to its limit.

Transcript

00:01 Solve this question as we are given that the radius of semicircle circle is equal to as it is is equal to 1.

00:15 So for this we have to draw a trapezium whose parallel sides are b1 and b2 respectively.

00:27 So for b1 is the bottom side and that is twice of r, two times of radius.

00:34 So we will take b1 as 2r.

00:37 So let's see our diagram first.

00:39 So here is the diagram and this is given as r is equal to 1.

00:43 So this side is b1 and above we have b2.

00:48 This angle is given as theta and when we join these sides we would get pi by 2 over.

00:56 So first of all we will draw perpendiculars from the vertex like this.

01:01 So in this way again we have to draw perpendicular from the center of the circle, then we will join these two also.

01:13 So total line segments are here.

01:15 And now when we find our area of trapeziate, area of traversiae, which is a is equal to 1 by 2, multiplied by height, sum of parallel sides.

01:29 So here b1 and b2.

01:33 So as per the values, we have here b1, which is equal to 2r.

01:39 So by using pythagoras theorem, we will find find our length for the radius and r square will be equal to h square plus b square by two whole square high partners square is equal to perpendicular square plus base square and it will be as h square added by b2 whole square divided by four which is equal to r square so when we take lcum and cross multiply both the sides so four r square is equal to four h squared plus b2 square so therefore b2 square will be equal to 4 r square minus 4 h square which is equal to 4 can be taken common r square plus minus h square so our value for b2 will be equal to within the square root we have to put whole value so through r square minus h square is the value of b2 so then after we can find our area of prepousiite or the trapezean which is equal to a is equal to 1 by 2 multiplied by height into b1 which is 2 r and b2 which is 2 times of square root r square minus h is square so for maximum area maximum area d a by b h is equal to 0 and therefore we can put here d a by b h t h, which is equal to r plus square root...

Question

Consider the sequence described by xn = 1/(1+n), n = 0, 1, 2, ... a) Find the suitable function g(x) so that the sequence can be generated by means of repeated substitution in the form xn+1 = g(xn), n = 0, 1, 2, ... b) Determine the rate of convergence of the sequence to its limit.

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