Why is the condition $n \ge 3$ required for the scaling bounds $n^2/2$ and $2n^2/3$ ?

Q: Why is the condition $n \ge 3$ required for the scaling bounds $n^2/2$ and $2n^2/3$?

The inequalities $n^2 - 3 \ge \frac{n^2}{2}$ and $n^2 - 3 \ge \frac{2n^2}{3}$ are not true for all $n$. They rely on $n$ being sufficiently large. Specifically, $n^2 - 3 \ge \frac{n^2}{2}$ simplifies to $\frac{n^2}{2} \ge 3 \implies n^2 \ge 6 \implies n \ge \sqrt{6} \approx 2.45$. Since $n$ is an integer, $n \ge 3$. Similarly, $n^2 - 3 \ge \frac{2n^2}{3}$ simplifies to $\frac{n^2}{3} \ge 3 \implies n^2 \ge 9 \implies n \ge 3$. Thus, $n \ge 3$ ensures the denominators are bounded below by the scaled terms, making the upper bounds for the error valid.

The inequalities

n^2 - 3 \ge \frac{n^2}{2}

and

n^2 - 3 \ge \frac{2n^2}{3}

are not true for all

n

. They rely on

n

being sufficiently large. Specifically,

n^2 - 3 \ge \frac{n^2}{2}

simplifies to

\frac{n^2}{2} \ge 3 \implies n^2 \ge 6 \implies n \ge \sqrt{6} \approx 2.45

. Since

n

is an integer,

n \ge 3

. Similarly,

n^2 - 3 \ge \frac{2n^2}{3}

simplifies to

\frac{n^2}{3} \ge 3 \implies n^2 \ge 9 \implies n \ge 3

. Thus,

n \ge 3

ensures the denominators are bounded below by the scaled terms, making the upper bounds for the error valid.

Conditions

Using scaling techniques for the error term $\frac{9}{n^2-3}$ .
$n$ is a positive integer.

Reasoning, step by step

Analyze the first scaling: $n^2 - 3 \ge \frac{n^2}{2}$ .
Rearrange: $n^2 - \frac{n^2}{2} \ge 3 \implies \frac{n^2}{2} \ge 3 \implies n^2 \ge 6$ .
Since $n$ is integer, $n \ge 3$ (as $2^2=4 < 6$ ).
Analyze the second scaling: $n^2 - 3 \ge \frac{2n^2}{3}$ .
Rearrange: $n^2 - \frac{2n^2}{3} \ge 3 \implies \frac{n^2}{3} \ge 3 \implies n^2 \ge 9$ .
Since $n$ is integer, $n \ge 3$ .
Conclude that for $n < 3$ , these specific lower bounds for the denominator do not hold, so the corresponding upper bounds for the error are invalid.

Example

The script explicitly states 'For $n\ge 3$ ' before applying both scaling methods. If $n=2$ , $n^2-3=1$ , but $n^2/2 = 2$ , and $1 \not\ge 2$ .

Common misconceptions

Assuming the scaling inequalities hold for all $n > \sqrt{3}$ .
Forgetting to check if the calculated $N$ satisfies the precondition $n \ge 3$ .
Believing that scaling always simplifies the problem without introducing additional constraints.

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Connected concepts

Limits →

Why is the condition $n \ge 3$ required for the scaling bounds $n^2/2$ and $2n^2/3$ ?

Conditions

Reasoning, step by step

Example

Common misconceptions

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Conditions

Reasoning, step by step

Example

Common misconceptions

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Why is the condition $n \ge 3$ required for the scaling bounds $n^2/2$ and $2n^2/3$ ?