For which x=x(n) does it hold that c^x = o((1)/(n}) where c in (0,1) is a constant. So clearly, for x=n, this is true. But for which x=o(n) does this hold? I thought of solving the equation c^x=1/n, which yields x=-(log n)/(log c).

ebendasqc

ebendasqc

Answered question

2022-10-19

Find real-valued sequences x ( n ) for which c x ( n ) = o ( 1 / n )
For which x = x ( n ) does it hold that
c x = o ( 1 n )
where c ( 0 , 1 ) is a constant.
So clearly, for x = n, this is true. But for which x = o ( n ) does this hold?
I thought of solving the equation c x = 1 n , which yields
x = log n log c .
But somehow, I am not sure. What does this mean? For x log n, we have that c x = o ( 1 n ) ?
And further: For which x = x ( n ) does it hold that
i = x n c i = o ( 1 n )

Answer & Explanation

n8ar1val

n8ar1val

Beginner2022-10-20Added 12 answers

Edited
1.Since log ( c ) < 0 we have x = 1 / o ( 1 / log ( n ) )
2.For fixed const c assuming x is integer this sum is c x c n 1 c . Since c n = o ( 1 / n ) the condition for x remains the same.

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