about infinite series.?

Question

(Zeno's Paradox) Achilles (A) and a tortoise (T) have a race. T gets a 1000 ft head start, but A runs at 10 ft/sec, whereas T only does .01 ft/sec. when A reaches T's starting pt, T has moved a short distance ahead, etc. Zeno claimed that A would never catch T. show that this is not so.

Answer 1

d = rt
d = 10 t for Achilles
(d - 1,000) = 0.1 t for the tortoise
t is the same in both cases
d/10 = (d - 1,000)/0.1
0.1d = 10d - 10,000
9.9d = 10,000
d = 1010.101 ft when Achilles catches the tortoise.
t = 101.0101 s

1,000/10 = 100
100*0.1 = 10, 1,000 + 10 = 1,010
10/10 = 1, 100 + 1 = 101
1*0.1 = 0.1,
etc.

You can set up a distance series of
d = 1000 + 10 + 0.1 + 0.01 + . . . . =
1,000(1 + 1/100 + (1/100)^2 + 1/100)^3 =
. . . . . n
1,000∑1/100^i = 1,000(1 - (1/100)^n)/(1 - 1/100)
. . . . i=0
lim[1,000(1 - (1/100)^n)/(1 - 1/100)] = 1,000/0.99
n→∞
= 1010.101010101010101010101010101 . . .
That this is an infinitely repeating decimal matters not. At t = 101.02 s
d = 10*101.02 = 1,010.2 for Achilles
(d - 1,000) = 0.1*101.02 = 10.102
d = 1,010.102 for the tortoise, and Achilles will have passed the tortoise.

Answer 2

The speed, expressed as a derivative, has a finite real limit.