- #1
Saitama
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Homework Statement
Let ##\displaystyle a_n=\frac 1 2+\frac 1 3+...+\frac 1 n##. Then
A)##a_n## is less than ##\displaystyle \int_2^n\frac{dx}{x}##.
B)##a_n## is greater than ##\displaystyle \int_1^n\frac{dx}{x}##.
C)##\displaystyle \lim_{n\rightarrow \infty} \frac{a_n}{\ln n}=1##
D)##\displaystyle \lim_{n\rightarrow \infty} a_n## is finite.
Homework Equations
The Attempt at a Solution
To my knowledge, there is no known closed form for the given ##a_n##.
I am clueless about the right approach so I started with ##n=2##. For n=2, ##a_2=0.5##
Also,
$$\int_2^2 \frac{dx}{x}=0$$
and
$$\int_1^2 \frac{dx}{x}=\ln 2 \approx 0.693$$
Obviously, A and B are not the answers.
How do I check for other options?
Any help is appreciated. Thanks!