FILIPE MOURA
(notice that
).
If we apply the formula for r=1, we get
.
In this case,
Using the formula from d,
Sn is increasing because
.
Sn is bounded above:
.
Therefore, Sn must converge.
Clearly, Rn>0.
We have
S2n=1, S2n+1=0. This implies that
does not exist.
After the first bounce, the ball rebounds to a heigth of
feet. The ball then falls from this heigth and rebounds
to
feet, etc.
Therefore, the total distance the ball travels is
If the interval
is divided into n equal
subintervals, and the left endpoint of the jth interval is
frac1j. Therefore,
.
Since
is a decreasing function on the interval
,
Ln > IN, for all
.
Since
,
and
.
For ,
.
This implies
.
Since
converges, so does
Since
,
Sn has the
desired accuracy if
.
A good estimate for the limit is
Since ak >0,
.
We have, for k>3,
,
or
.
Therefore,
converges,
since
converges to
.
We have
Therefore, by the ratio test, the series converges.
(sum of two absolutely convergent series).