.
(I) At small values of real positive quantity x, ln(1+x) is equivalent to x
as a small quantity in Calculus.
Even more precisely, 0 < <= x
THEREFORE, 0 < <= .
The series is convergent, which is well known fact in Calculus after Euler.
So, the series is convergent, too.
(II) When x is small, sin(x) is equivalent to x as a small quantity in Calculus.
Therefore, is equivalent to .
The series is harmonic series, and it is divergent - which is
a well known fact in Calculus.
It implies that the series is divergent, too.
(III) Estimate = is equivalent to n^(3/2)*(1/(2n^3)) = (1/2)*(1/n^(3/2)).
Therefore, it is OBVIOUS that the series is convergent.
Thus we have this conclusion: series (I) converges; series (II) diverges; series (III) converges.
ANSWER. Option (B).
Solved.
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Comment from student: Thank you. I am confused on the step that you explained (I took a screenshot of it):
https://drive.google.com/file/d/1SMF_1rbZAQCY0SzU_9WfjtzSff8WLDbC/view?usp=sharing
How did you manipulate algebraically?
My response: the question is to explain why
is equivalent to . (*)
Notice that in this problem n is a BIG value; so, and are small quantities.
In Calculus, if " a " is a small value, then
is equivalent, in order of smallness, to . (**)
It is from the area of common knowledge in Calculus.
So, our estimate (*) is a direct consequense of (**).
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If you seriously want to understand the solutions of this problem and other similar problems
from your list, you should be FREE in manipulating basic knowledge of Calculus,
since they are PRE-REQUIZITES for solving such problems.