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Exercises relating to sequences.

The sequence $a_n = 1/n^3$ has limit $L = 0$. Suppose $\epsilon = 1/64$; find a threshold $N$ such that is guaranteed to hold for all $n > N$ (take your value of $N$ as small as possible).
The sequence $a_n = (2n^2 + (-1)^n)/n^2$ has limit $L = 2$. Suppose $\epsilon = 1/100$; find a threshold $N$ such that is guaranteed to hold for all $n > N$ (take your value of $N$ as small as possible).
Determine the $n^\text {th}$ term of the given sequence. $a_1 = 4$, $a_2 = 7$, $a_3 = 10$, $a_4 = 13$, $a_5 = 16$, $\ldots$
Determine the $n^\text {th}$ term of the given sequence. $a_1 = 3$, $a_2 = -5/2$, $a_3 = 7/4$, $a_4 = -9/8$, $a_5 = 11/16$, $\ldots$
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {N/A}$ (enter N/A if the sequence does not converge to a finite answer).
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {4/3}$ (enter N/A if the sequence does not converge to a finite answer).
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {0}$ (enter N/A if the sequence does not converge to a finite answer).
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {e^3}$ (enter N/A if the sequence does not converge to a finite answer).
Take the reciprocal and compare to your reference list of commonly-occurring limits.
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {N/A}$ (enter N/A if the sequence does not converge to a finite answer).
If a sequence $b_n$ tends to $e^3$, what will the sequence $(b_n)^n$ do?
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {N/A}$ (enter N/A if the sequence does not converge to a finite answer).
What are the relative orders of growth of numerator versus denominator?
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {0}$ (enter N/A if the sequence does not converge to a finite answer).
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {0}$ (enter N/A if the sequence does not converge to a finite answer).
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {0}$ (enter N/A if the sequence does not converge to a finite answer).
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {0}$ (enter N/A if the sequence does not converge to a finite answer).
Let $b_n$ be the sequence given by converges. Compute its limit.
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {N/A}$ (enter N/A if the sequence does not converge to a finite answer).
Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. The sequence convergesdiverges to $\answer {2}$ (enter N/A if the sequence does not converge to a finite answer).
Let $a_n$ be the sequence given by converges. Compute its limit.
If $a_n$ happens to be positive and less than $1/2$, then $2 (1-a_n) > 1$, so this forces $2 a_n(1-a_n) > a_n$ (meaning that the term after $a_n$ will be larger than $a_n$.
The function $2 x(1-x)$ is nonnegative on the interval $[0,1]$ and has a maximum value of $1/2$ attained at $x = 1/2$. This means that if $a_n$ is anything between $0$ and $1$, the next term of the sequence will always be between $0$ and $1/2$.

### Sample Quiz Questions

Find the limit of the sequence Justify your response. (Hints will not be revealed until after you choose a response.)

$\displaystyle 0$ $\displaystyle \frac {1}{3}$ $\displaystyle \frac {1}{2}$ $\displaystyle 1$ $\displaystyle 2$ $\displaystyle 3$

Determine whether the limit below exists. If it exists, find its value. Justify your response. (Hints will not be revealed until after you choose a response.)

$\displaystyle -1$ $\displaystyle 0$ $\displaystyle \frac {1}{2}$ $\displaystyle 2$ $\displaystyle 3$ limit does not exist

Determine whether the limit below exists. If it exists, find its value. Justify your response. (Hints will not be revealed until after you choose a response.)

$\displaystyle 0$ $\displaystyle 1$ $\displaystyle e^{-1}$ $\displaystyle e$ $\displaystyle e^2$ limit does not exist

Determine whether the limit below exists. If it exists, find its value. Justify your response. (Hints will not be revealed until after you choose a response.)

$\displaystyle 0$ $\displaystyle 1$ $\displaystyle e^{-1}$ $\displaystyle e$ $\displaystyle e^2$ limit does not exist

### Sample Exam Questions

Determine whether the sequence $\displaystyle a_n = (-1)^{n-1} \frac {n^2}{1 + n^2 + n^3}$ converges or diverges. If it converges, find its limit.

divergent, $\displaystyle \lim _{n \rightarrow \infty } a_n = 0$ convergent, $\displaystyle \lim _{n \rightarrow \infty } a_n = 1$ convergent, $\displaystyle \lim _{n \rightarrow \infty } a_n = 0$ convergent, $\displaystyle \lim _{n \rightarrow \infty } a_n = -1$ divergent, $\displaystyle \lim _{n \rightarrow \infty } a_n = \infty$ divergent, limit doesn’t exist