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For  $$\alpha \geq 0$$​ , define

​$$a_n = \frac{1 + 2^\alpha + \cdots + n^\alpha}{n^{\alpha+1}}.$$​​

What is the value of  $$\lim\limits_{n \to \infty} a_n$$​ ?

$$\lim_{n \to \infty} \frac{((n+1)(n+2) \cdots (n+n))^{1/n}}{n}.$$​

Let $\phi (n)$ be the cardinality of the set:

​$$\{a\mid1\leq a \leq n, gcd(a,n)=1\}$$

Where $\gcd (a,n)$ denotes the greatest common divisor of $a$ and $n$.

Which of the following is NOT true?​​

{a∣1≤a≤n,gcd⁡(a,n)=1}\{ a \mid 1 \leq a \leq n, \, \gcd(a, n) = 1 \}

For a positive integer n, let $$f^{(n)}$$​ denote the n-th derivative of f.

Suppose anentire function f satisfies: $$f^{(2)}+f=0$$​​

Which of the following is correct?

Let X, Y be defined by

​$$X = \{(x_n)_{n \geq 1} : \limsup_{n \to \infty} x_n = 1, \; \text{where} \; x_n \in \{0, 1\}\}$$​​

and

​$$Y = \{(x_n)_{n \geq 1} : \lim_{n \to \infty} x_n \; \text{does not exist, where} \; x_n \in \{0, 1\}\}$$​.

Which of the following is true?

Suppose $$(a_n)_{n \geq 1} \ and\ (b_n)_{n \geq 1}$$​ are two bounded sequences of real numbers.
Which of the following is true?

Consider the series $$\sum_{n=3}^\infty \frac{a^n}{n^b (\log_e n)^c}.$$​​

For which values of $$a, b, c \in \mathbb{R}$$​, does the series $$\textbf{NOT converge}?$$​​

Let  $$a_n = n + n^{-1}$$​. Which of the following is true for the series:
​$$\sum_{n=1}^\infty \frac{(-1)^{n+1} a_{n+1}}{n!}$$ ?

​$$\text{Given } (a_n)_{n \geq 1} \text{ a sequence of real numbers}\\ \text{, which of the following statements is true?}$$​​

​$$f:\N\to\N\ \text{be a bounded function.Which of the}\\\text{following statements is not true?}$$​​