The discussion revolves around the Direct Comparison Test in the context of inequalities involving logarithmic and polynomial functions, specifically the inequality ln(n) < n^(1/10). Participants are exploring the validity of this inequality for various values of n and its implications for the test.
The discussion is ongoing, with participants providing differing perspectives on the inequality's validity. Some have pointed out specific values of n where the inequality holds, while others are questioning the assumptions behind the inequality's application in the context of the Direct Comparison Test.
There is a noted focus on the conditions under which the inequality ln(n) < n^(1/10) is true, with some participants emphasizing the need for justification over an infinitely long interval rather than just specific cases.
And what did you find out?dami said:Homework Statement
Explain why the Direct Comparison Test allows us to use the inequality Ln n < n^(1/10) even though it is not true for a great many n values.
Homework Equations
The Attempt at a Solution
I looked at the graphs of Ln (n) vs. n^(k)
That is incorrect. The statement ln n < n.1 is true for n = 3, and it is also true for a lot of much larger values.dami said:Actually, ln n < n^{1/10} is only true for n < 3 (for integer values of n).