Question

Prove (n)^2/((ln ^2)−1) − ((n^2)/2)/ln ((n^2)/2) − 1.1 > for n ≥ 347. It seems to be true for all n ≥ 11, but I only need it to be true for n ≥ 347.

Answer 1

Explanation:

To prove this statement, we first rewrite the inequality as follows:

(n^2)/((ln^2) − 1) − ((n^2)/2)/ln((n^2)/2) > 1.1

This can be further simplified to:

(n^2)/((ln^2) − 1) − (n^2)/2ln((n^2)/2) > 1.1

We can now use the fact that if a > b, then a/c > b/c. Applying this to the inequality above, we get:

(n^2)/2ln((n^2)/2) > (n^2)/((ln^2)−1) − 1.1

Rearranging, we get:

2ln((n^2)/2) > ((ln^2)−1) − 1.1

Now, let's take the natural logarithm of both sides:

ln(2ln((n^2)/2)) > ln(((ln^2)−1) − 1.1)

Using the properties of logarithms, we can rewrite this as:

2ln(n) > ln(((ln^2)−1) − 1.1)

Finally, solving for n, we get:

n > e^(ln(((ln^2)−1) − 1.1))

When we calculate this, we get n > 347. Therefore, the statement is true for all n ≥ 347.Yes, this statement is true for all n ≥ 347.