Question:

Does the sequence{an} with an=nsin(
1

n
) converge or diverge? If it converges, find it's limit.

A student hands in the following solution. Is it correct?

Line 1: One way to determine if a sequence converges is to try to evaluate the limit as n⟶∞ .

Line 2: lim
n→∞an=lim
n→∞nsin(
1

n
)=

Line 3: lim
n→∞
sin(
1

n
)

1

n
=

Line 4: Both numerator and denominator approach 0 as n⟶∞ , so we can use l'Hopital's rule:

Line 5: lim
n→∞
cos(n)(−
1

n2
)

−
1

n2
=

Line 6: lim
n→∞cos(n)=∞

Line 7: Because the limit is ∞, this means the sequence diverges.

Is the student's solution correct? If not, in which line is the first mistake? Incorrect, with first error in line 5

If not, what is the correct final answer to the question? Final answer: The sequence converges to 1

Question

Question:

Does the sequence{an}  with an=nsin(
1

n
)  converge or diverge? If it converges, find it's limit.

A student hands in the following solution. Is it correct?

Line 1: One way to determine if a sequence converges is to try to evaluate the limit as n⟶∞ .

Line 2: lim
n→∞an=lim
n→∞nsin(
1

n
)=

Line 3: lim
n→∞
sin(
1

n
)

1

n
=

Line 4: Both numerator and denominator approach 0 as n⟶∞ , so we can use l'Hopital's rule:

Line 5: lim
n→∞
cos(n)(−
1

n2
)

−
1

n2
=

Line 6: lim
n→∞cos(n)=∞

Line 7: Because the limit is ∞, this means the sequence diverges.

Is the student's solution correct? If not, in which line is the first mistake?  Incorrect, with first error in line 5

If not, what is the correct final answer to the question? Final answer: The sequence converges to 1

BlackTom AI · Accepted Answer

The prompt presents a sequence a_n = n sin(1/n) and asks about convergence, plus a student solution with two listed conclusions. We will evaluate each provided option carefully and point out the reasoning issues in the student’s steps before confirming the true behavior of the sequence.

Option 1: "Incorrect, with first error in line 5"
- The claim here is that the first incorrect line occurs at line 5. To judge this, we examine the student’s steps up to line 5. The student rewrites the limit as lim_{n→∞} [n sin(1/n)]. They then consider lim sin(1/n)/(1/n), which is a valid reformulation since a_n = n * sin(1/n) = sin(1/n) / (1/n). The observation that both numerator sin(1/n) and denominator (1/n) tend to 0 as n→∞ is correct, so the 0/0 indeterminate form is identified appropriately.
- The first genuine mathematical misstep appears at line 5, where the student attempts to apply L'Hôpital's rule to a limit with respect to the discrete variable n. L'Hôpital’s rule is formulated for limits of real-valued functions as a real variable x → a, not for sequences indexed by integers n. Moreover, the derivative shown in line 5 is incorrect: the derivative of sin(1/n) with respect to n is cos(1/n) * (-1/n^2), not cos(n) * (-1/n^2). This indicates a misapplication and a factual error in differentiation.
- Given these points, identifying line 5 as the first error is justified, since the misuse of L'Hôpital’s rule and the incorrect derivative both arise there, and no earlier line contains a faux step of equal or greater severity.
- The reasoning here would be strengthened by noting a more natural limit evaluation: using the standard limit lim_{x→0} sin x / x = 1, with x = 1/n, to conclude lim_{n→∞} sin(1/n) / (1/n) = 1, which directly yields lim_{n→∞} a_n = 1 without invoking L'Hôpital’s rule.

Option 2: "Final answer: The sequence converges to 1"
- This option asserts the final value of the limit is 1. Indeed, since a_n = n sin(1/n) and using the standard limit sin x ~ x as x → 0 (or equivalently lim_{x→0} sin x / x = 1 with x = 1/n), we obtain lim_{n→∞} sin(1/n) / (1/n) = 1. Therefore, lim_{n→∞} a_n = 1, so the sequence converges and its limit is 1.
- The core conceptual point here is the continuity of the sine function near 0 and the small-angle approximation sin t ≈ t for t near 0, which justifies evaluating sin(1/n) as approximately 1/n for large n. This produces a clean limit of 1 for the sequence.

Synthesis of the analysis across options:
- The first option correctly identifies that there is an error in the student’s reasoning, and pinpoints the first incorrect step at line 5 where L'Hôpital’s rule is misapplied to a discrete-index limit and the derivative is written incorrectly. The identification of line 5 as the first mistake is defensible given the sequence-context mismatch and the derivative mistake demonstrated.
- The second option makes a true mathematical claim about the limit of the sequence. The conclusion that the sequence converges to 1 matches the established limit obtained via the standard sin x / x limit with x = 1/n.
- The overall explanation should be careful to separate the evaluation of the limit (which is correct) from the critique of the student’s method (where the error occurs). It is important to emphasize that while the final limit is 1, the justification in line 5 is invalid due to the inappropriate use of L'Hôpital’s rule for sequences and the incorrect derivative expression.

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