Notice that
So as
you have
. Clearly
must converge.
The second sequence requires a bit more work.
The monotone convergence theorem will help here; if we can show that the sequence is monotonic and bounded, then
will converge.
Monotonicity is often easier to establish IMO. You can do so by induction. When
, you have
Assume
, i.e. that
. Then for
, you have
which suggests that for all
, you have
, so the sequence is increasing monotonically.
Next, based on the fact that both
and
, a reasonable guess for an upper bound may be 2. Let's convince ourselves that this is the case first by example, then by proof.
We have
and so on. We're getting an inkling that the explicit closed form for the sequence may be
, but that's not what's asked for here. At any rate, it appears reasonable that the exponent will steadily approach 1. Let's prove this.
Clearly,
. Let's assume this is the case for
, i.e. that
. Now for
, we have
and so by induction, it follows that
for all
.
Therefore the second sequence must also converge (to 2).
Numbers on the y-axis on a coordinate plane increase in UP direction.
Left hand side is '-x', right hand side is 'x'
Up is 'y' and down is '-y'
Y is the vertical axis and X is horizontal axis.
Answer:
a) the number of miles in 1 kilometer
Step-by-step explanation:
The car converts from miles/hour to kilometers/hour, if we see the time measurement (hours) it stays the same in both units.
So to make the conversion<u> it is enough to know how many miles are in one kilometer.</u>
For example, lets convert 10miles/hour to kilometers/hour.
there are 0.621371 miles in 1 kilometer, so if we divide 10miles/hour by 0.621371 we get kilometers/hour units:
Thus to make the conversion between the two units is needed the number of miles in 1 kilometer.
Answer Step-by-step explanation: