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CHAPTER 36
36.11
0.9, 0.99, 0.999, 0.9999,
36.12
36.13
36.14
36.15
36.16
36.17
36.18
36.19
36.20
36.21
For
Since
Let K be the least integer
For
Each of
Therefore,
as
Hence,
Here we have used the fact that
which follows by
L'Hopital's rule.
COS 77, COS (7T/2), COS (7T/3), COS (17/4), ....
In Problems 36.19-36.45, determine whether the given sequence converges, and, if it does, find the limit.
a n = cos (ir/n) —» cos 0 = 1.
The sequence takes on the values V2/2,1, V2/2,0, -V2/2, -1, - V2/2,0, and then keeps repeating in this
manner. Hence, there is no limit.
As L'Hopital's rule yields
The sequence converges to 0 (see Problem 36.15).
So
a n = sin(rt7r/4).
a n = nle".
a n = (Inn) In.