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).
The Direct Comparison Test inequality is a method used to determine whether an infinite series converges or diverges. It compares the given series to a known convergent or divergent series, and if the known series is larger than the given series, then the given series must also converge. This test is only applicable for series with positive terms.
The Direct Comparison Test inequality is most useful when the terms of the given series are not easily compared to a known series, such as a geometric or p-series. It can also be used when the Limit Comparison Test is inconclusive.
The steps for using the Direct Comparison Test inequality are as follows:
No, the Direct Comparison Test inequality can only be used for series with positive terms. If the given series has negative terms, the Absolute Comparison Test must be used instead.
Yes, the Direct Comparison Test inequality can only be used for series with positive terms and is only applicable when the known series is larger than the given series. It also cannot be used for alternating series or series with terms that do not approach zero.