Evaluate $\lim_{n \rightarrow \infty} \left\{\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{n+n}\right\}$.

Let $\{x_n\}$ be a sequence such that $x_1 = 2$, $x_2 = 1$ and $2x_n - 3x_{n-1} + x_{n-2} = 0$ for $n > 2$. Find an expression for $x_n$.

Let $f(n)$ satisfy the recurrence $f(n) + f(n-1) = nf(n-1) + (n-1)f(n-2)$ with $f(0) = 1$, $f(1) = 0$. Find a closed form for $f(n)$.

Find the general term $T_r$ of the sequence $2, 7, 14, 23, 34, \ldots$

For any real number $x$, let $\tan^{-1}(x)$ denote the unique real number $\theta$ in $(-\pi/2, \pi/2)$ such that $\tan\theta = x$. Then $\lim_{n \to \infty} \sum_{m=1}^{n} \tan^{-1}\left\{\frac{1}{1+m+m^{2}}\right\}$
(a) Is equal to $\pi/2$
(b) Is equal to $\pi/4$
(c) Does not exist
(d) None of the above.

Find $\lim_{n \to \infty} u_n$ where $u_n = \dfrac{1}{2} + \dfrac{2}{2^2} + \dfrac{3}{2^3} + \dfrac{4}{2^4} + \cdots + \dfrac{n}{2^n}$.

Let $f$ be a periodic function with period $1$, and let $g(t) = \int_0^t f(x)\,dx$. Define $h(t) = \lim_{n\to\infty} \dfrac{g(t+n)}{n}$. Which of the following is true about $h(t)$?
(A) $h(t)$ depends on $t$
(B) $h(t)$ is not defined for all $t$
(C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$
(D) None of the above

Let $a_1 = 24^{1/3}$ and $a_{n+1} = (a_n + 24)^{1/3}$. Find the integer part of $a_{100}$.

Let $x_n = \dfrac{1}{(2a)^n}$ for $a > 0$. Which of the following is true?
(A) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$
(B) $x_n \to 0$ if $0 < a < 1/2$ and $x_n \to \infty$ if $a > 1/2$
(C) $x_n \to 1$ for all $a > 0$
(D) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$

Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then,
(A) $a _ { n } < b _ { n }$ for all $n > 1$
(B) $a _ { n } > b _ { n }$ for all $n > 1$
(C) $a _ { n } = b _ { n }$ for infinitely many $n$
(D) None of the above

If $a _ { n } = \left( 1 + \frac { 1 } { n ^ { 2 } } \right) \left( 1 + \frac { 2 ^ { 2 } } { n ^ { 2 } } \right) ^ { 2 } \left( 1 + \frac { 3 ^ { 2 } } { n ^ { 2 } } \right) ^ { 3 } \cdots \left( 1 + \frac { n ^ { 2 } } { n ^ { 2 } } \right) ^ { n }$, then $$\lim _ { n \rightarrow \infty } a _ { n } ^ { - 1 / n ^ { 2 } }$$ is
(A) 0
(B) 1
(C) $e$
(D) $\sqrt { e } / 2$

For any $n \geq 5$, the value of $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{2^n - 1}$ lies between
(A) 0 and $\frac{n}{2}$
(B) $\frac{n}{2}$ and $n$
(C) $n$ and $2n$
(D) none of the above.

The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is:
(A) $\frac{3}{4}$
(B) $\frac{1}{4}$
(C) 1
(D) 4

For any integer $n \geq 1$, define $a_n = \frac{1000^n}{n!}$. Then the sequence $\{ a_n \}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

Find the maximum among $1,2 ^ { 1/2 } , 3 ^ { 1/3 } , 4 ^ { 1/4 } , \ldots$.

Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then,
(a) $a _ { n } < b _ { n }$ for all $n > 1$
(b) $a _ { n } > b _ { n }$ for all $n > 1$
(c) $a _ { n } = b _ { n }$ for infinitely many $n$
(d) none of the above.

Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then,
(a) $a _ { n } < b _ { n }$ for all $n > 1$
(b) $a _ { n } > b _ { n }$ for all $n > 1$
(c) $a _ { n } = b _ { n }$ for infinitely many $n$
(d) none of the above.

The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is:
(a) $\frac { 3 } { 4 }$
(b) $\frac { 1 } { 4 }$
(c) 1
(d) 4.

The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is:
(a) $\frac { 3 } { 4 }$
(b) $\frac { 1 } { 4 }$
(c) 1
(d) 4.

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(a) does not have a maximum
(b) attains maximum at exactly one value of $n$
(c) attains maximum at exactly two values of $n$
(d) attains maximum for infinitely many values of $n$.

Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then,
(A) $a _ { n } < b _ { n }$ for all $n > 1$
(B) $a _ { n } > b _ { n }$ for all $n > 1$
(C) $a _ { n } = b _ { n }$ for infinitely many $n$
(D) None of the above

If $a _ { n } = \left( 1 + \frac { 1 } { n ^ { 2 } } \right) \left( 1 + \frac { 2 ^ { 2 } } { n ^ { 2 } } \right) ^ { 2 } \left( 1 + \frac { 3 ^ { 2 } } { n ^ { 2 } } \right) ^ { 3 } \cdots \left( 1 + \frac { n ^ { 2 } } { n ^ { 2 } } \right) ^ { n }$, then $$\lim _ { n \rightarrow \infty } a _ { n } ^ { - 1 / n ^ { 2 } }$$ is
(A) 0
(B) 1
(C) $e$
(D) $\sqrt { e } / 2$

If $a _ { n } = \left( 1 + \frac { 1 } { n ^ { 2 } } \right) \left( 1 + \frac { 2 ^ { 2 } } { n ^ { 2 } } \right) ^ { 2 } \left( 1 + \frac { 3 ^ { 2 } } { n ^ { 2 } } \right) ^ { 3 } \cdots \left( 1 + \frac { n ^ { 2 } } { n ^ { 2 } } \right) ^ { n }$, then $$\lim _ { n \rightarrow \infty } a _ { n } ^ { - 1 / n ^ { 2 } }$$ is
(A) 0
(B) 1
(C) $e$
(D) $\sqrt { e } / 2$

The value of $\lim_{n \rightarrow \infty} \frac{1^3 + 2^3 + \ldots + n^3}{n^4}$ is:
(A) $\frac{3}{4}$
(B) $\frac{1}{4}$
(C) 1
(D) 4

The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is:
(A) $\frac { 3 } { 4 }$
(B) $\frac { 1 } { 4 }$
(C) 1
(D) 4