Define a sequence An=m+Un−1 of which all the terms are clearly irrational since U_n is irrational for all n and a rational number + irrational number is always irrational. Yet it converges to m as n→∞
Is this what you mean?
Yes. I doubt it would lose you much here, because it's so trivial to fix it to work for the general case. But the standard interpretation of "show XXX for a given integer n" means "it has to work for any n, but you can consider n as fixed".
Yes. I doubt it would lose you much here, because it's so trivial to fix it to work for the general case. But the standard interpretation of "show XXX for a given integer n" means "it has to work for any n, but you can consider n as fixed".
To find the area in the three listed cases, we note that i) and ii) are both specific versions of iii), therefore we need only find the answer to iii) and i) and ii) follow easily.
From a diagram, we see that we must replace d with the perpendicular distance from the line to the furthest point on the circle from the line. This gives us:
d+D=R, where D is the shortest distance from the origin to the line. The distance from a point (x, y) to the centre is given by
D2=(x−a)2+(y−b)2
But the point (x, y) lies on the line y = mx + c, therefore
D2=(x−a)2+(mx+c−b)2
To find the minimum value of D, we must differentiate this expression and set it equal to 0. This gives us
x−a+m2x+mc−mb=0
⟹x=m2+1mb−mc+a
substituting this back into our expression for D^2, we get
D2=(m2+1)2(mb−mc−m2a)2+(ma+c−b)2
and therefore, to find the new area, we must modify (*) such that
d=R−(m2+1)2(mb−mc−m2a)2+(ma+c−b)2
The answer to i) follows by letting m = 0; the answer to ii) follows by letting a = b = 0.
I grant you it's not 100% clear, but I would interpret this part as "given an integer m, give an example of a sequence of irrational numbers that converges to m". (In other words, you have to produce a sequence that works for every m).
Sorry to pester you, but when it says "converges to a given integer m", does that mean that its sum converges to m or does that mean the general term tends to m?
Sorry to pester you, but when it says "converges to a given integer m", does that mean that its sum converges to m or does that mean the general term tends to m?
If a sequence converges to a value, then the general term converges to that value If a series or sum converges to a value then the sum of the series converges to that value