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Mathematical Expression Editor

Exercises relating to sequences.

The sequence has limit . Suppose ; find a threshold such that is guaranteed to hold
for all (take your value of as small as possible).

The sequence has limit . Suppose ; find a threshold such that is guaranteed to hold
for all (take your value of as small as possible).

Determine the term of the given sequence. , , , , ,

Determine the term of the given sequence. , , , , ,

Determine whether the sequence converges or diverges. If convergent, give the limit of
the sequence. The sequence convergesdiverges to (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 (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 (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 (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 (enter N/A if the sequence does not converge to a finite answer).

If a sequence
tends to , what will the sequence do?

Determine whether the sequence converges or diverges. If convergent, give the limit
of the sequence. The sequence convergesdiverges to (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 (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 (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 (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 (enter N/A if the sequence does not converge to a finite answer).

Let 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 (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 (enter N/A if the sequence does not converge to a finite answer).

Let be the sequence given by converges. Compute its limit.

If happens to
be positive and less than , then , so this forces (meaning that the term
after will be larger than .

The function is nonnegative on the interval
and has a maximum value of attained at . This means that if is anything
between and , the next term of the sequence will always be between and
.

Sample Quiz Questions

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

Because the square root function is continuous, you can pass the limit through it and
compute

Reduce numerator and denominator to the dominant terms (in the regime
).

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.)

limit does not exist

Comparing the orders of growth of the terms in the numerator, the first term
dominates because .

Likewise the first term dominates in the denominator because .

Neglecting non-dominant terms leads to the limit which simply equals
.

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.)

limit does not exist

First observe that as .

Next, in light of the known limit as , manipulate
exponents to see that

As , the first term on the right-hand side tends to
and the second term tends to . Thus the original sequence tends to as
well.

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.)

limit does not exist

First observe that as .

Since the limit is positive and less than one, raising this
expression to increasingly large powers generates a sequence which converges rapidly
to zero.

Sample Exam Questions

Determine whether the sequence converges or diverges. If it converges, find its limit.