2 to the power of negative 2
2 answers:
You might be interested in
Answer:
a) we know that this is convergent.
b) we know that this might not converge.
Step-by-step explanation:
Given the is convergent
Therefore,
(a) The power series
has radius of convergence at least as big as 8. So we definitely know it converges for all x satisfying -8<x≤8. In particular for x = -3
∴ is convergent.
(b) -8 could be right on the edge of the interval of convergence, and so might not converge
>= means greater than or equal to
<= means less than or equal to
---------------------------------------------
Part A
The graph of y >= -3x+3 will have a solid boundary line and the shading will be above the boundary line.
The boundary line y = -3x+3 has a negative slope so it moves down as you read it from left to right. It goes through the points (0,3) and (1,0)
--------------
The graph of y < (3/2)x - 6 will have a dashed or dotted boundary line. The shading is below the boundary.
The graph y = (3/2)x-6 goes through the two points (0,-6) and (2,-3)
--------------
If you graph both y >= -3x+3 and y < (3/2)x - 6 together, you get what you see in the attached image. This solution shaded region is the result of the overlapping prior shaded regions.
---------------------------------------------
Part B
Plug (x,y) = (-6,3) into each inequality to see if we get a true inequality or not
For the first inequality we have
y >= -3x+3
3 >= -3*(-6)+3
3 >= 18+3
3 >= 21
which is false. The value 3 is not larger or equal to 21. So right off the bat we know that (-6,3) is NOT a solution. It is NOT in the solution region.
Let's check the other inequality just for the sake of completeness
y < (3/2)x - 6
3 < (3/2)*(-6) - 6
3 < -9 - 6
3 < -15
this is also false. The value -15 is smaller than 3, since it is to the left of 3
We're given more evidence that (-6,3) is NOT in the solution area. It is outside of both shaded areas.
<= means less than or equal to
---------------------------------------------
Part A
The graph of y >= -3x+3 will have a solid boundary line and the shading will be above the boundary line.
The boundary line y = -3x+3 has a negative slope so it moves down as you read it from left to right. It goes through the points (0,3) and (1,0)
--------------
The graph of y < (3/2)x - 6 will have a dashed or dotted boundary line. The shading is below the boundary.
The graph y = (3/2)x-6 goes through the two points (0,-6) and (2,-3)
--------------
If you graph both y >= -3x+3 and y < (3/2)x - 6 together, you get what you see in the attached image. This solution shaded region is the result of the overlapping prior shaded regions.
---------------------------------------------
Part B
Plug (x,y) = (-6,3) into each inequality to see if we get a true inequality or not
For the first inequality we have
y >= -3x+3
3 >= -3*(-6)+3
3 >= 18+3
3 >= 21
which is false. The value 3 is not larger or equal to 21. So right off the bat we know that (-6,3) is NOT a solution. It is NOT in the solution region.
Let's check the other inequality just for the sake of completeness
y < (3/2)x - 6
3 < (3/2)*(-6) - 6
3 < -9 - 6
3 < -15
this is also false. The value -15 is smaller than 3, since it is to the left of 3
We're given more evidence that (-6,3) is NOT in the solution area. It is outside of both shaded areas.