What is lim inf sin((n*pi)/8) as n --> infinity.
is n an integer? Or any value?
n is an natural number.
To find the lim inf (limit inferior) of a sequence, we need to find the greatest lower bound of the values in the sequence. In this case, the sequence is given by sin((n*pi)/8), where n approaches infinity.
To approach this problem, we can observe that the value of sin(x) oscillates between -1 and 1 as x varies.
Let's calculate the values of sin((n*pi)/8) for the first few values of n:
n = 1: sin((pi)/8) ≈ 0.383
n = 2: sin((2*pi)/8) ≈ 0.707
n = 3: sin((3*pi)/8) ≈ 0.924
n = 4: sin((4*pi)/8) ≈ 1 (Note: sin((4*pi)/8) = sin((pi)/2) = 1)
n = 5: sin((5*pi)/8) ≈ 0.924
...
We can observe that the values of sin((n*pi)/8) are bounded between -1 and 1, and they repeat periodically. As n gets larger and larger, the values of sin((n*pi)/8) will cover all the values between -1 and 1 infinitely many times.
Since the values are bounded and repeat infinitely, the lim inf of sin((n*pi)/8) as n approaches infinity can be determined as the greatest lower bound of the sequence, which is -1.
Therefore, the lim inf of sin((n*pi)/8) as n approaches infinity is -1.