Prove the following inequality:

\(\displaystyle \left(1+\frac1n\right)^n\geq2.\)

Difficulty: 3.

Calculate the limit of the sequence

\(\displaystyle a_n=\left(\frac{n+2}{n+1}\right)^n .\)

Difficulty: 4. Solution is available for this problem.

Calculate:

\(\displaystyle \lim\left(\frac{n+3}{n-1}\right)^{3n+8} = ?\)

Difficulty: 4.

Prove the following inequalities:

\(\displaystyle \left({n\over e}\right)^n<n!<e\cdot\left({n\over 2}\right)^n.\)

Difficulty: 5.

Prove that

\(\displaystyle \left (1+\frac{1}{n} \right )^{n+1}>\left (1+\frac{1}{n+1} \right)^{n+2},\)

in other words the sequence \(\displaystyle a_n=\left (1+\frac{1}{n} \right )^{n+1}\) is strictly monotone decreasing.

Difficulty: 5. Solution is available for this problem.

Prove that

\(\displaystyle n+1 < e^{1+\frac{1}{2}+\ldots+\frac{1}{n}} < 3n. \)

Difficulty: 5.

Prove that for all \(\displaystyle n\in \N\) we have \(\displaystyle \displaystyle n!>\left(\frac{n+1}e\right)^n\), and for \(\displaystyle n\ge7\) we have \(\displaystyle \displaystyle n!<\frac{n^{n+1}}{e^n}\).

Difficulty: 5.

Which one is the greater? \(\displaystyle 1000001^{1000000}\) or \(\displaystyle 1000000^{1000001}\).

Difficulty: 6.

Prove the following inequalities.

\(\displaystyle 0<e-\left(1+{1\over n}\right)^n<{3\over n}.\)

Difficulty: 7.

Find positive constants \(\displaystyle c_1,c_2\) for which

\(\displaystyle c_1 \cdot \frac{n^{n+\frac12}}{e^n} < n! < c_2 \cdot \frac{n^{n+\frac12}}{e^n} \)

for all \(\displaystyle n\in \N\).

Difficulty: 7.

Verify that if \(\displaystyle n\cdot a_n\to a\) and \(\displaystyle b_n/n\to b\), then \(\displaystyle (1+a_n)^{b_n}\to e^{ab}\).

Difficulty: 7.

Prove for every sequence \(\displaystyle (a_n)\):

\(\displaystyle \liminf \left(1+\frac1n\right)^{a_n} = e^{\liminf\tfrac{a_n}{n}}. \)

Difficulty: 7.

Which one is greater? The number \(\displaystyle e\) or \(\displaystyle \displaystyle\left(1+\frac1n\right)^{n+\frac12}\)?

Difficulty: 9.