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It was shown in herehere that $$\left(1+\frac{1}{n}\right)^n < n$$ for $$n>3$$. I think we can be come up with a better bound, as follows:

$$\left(1+\frac{1}{n}\right)^n \le 3-\frac{1}{n}$$ for all natural number $$n$$.

The result is true for all real number $$\ge 1$$, which can be shown using calculus. I wonder if the above result can be proved using mathematical induction?

I have tried but fail! Anyway, this question is also inspired by, and related to this questionthis question.

Edit:

I also found that $$\left(1+\frac{1}{n+k}\right)^n \le 3-\frac{k+1}{n}$$ for all natural number $$k$$, some large $$N$$ and $$n > N$$. This implies that $$\left(1+\frac{1}{2n}\right)^n \le 2-\frac{1}{n}.$$

And again, I can't prove any of them using Mathematical Induction.

It was shown in here that $$\left(1+\frac{1}{n}\right)^n < n$$ for $$n>3$$. I think we can be come up with a better bound, as follows:

$$\left(1+\frac{1}{n}\right)^n \le 3-\frac{1}{n}$$ for all natural number $$n$$.

The result is true for all real number $$\ge 1$$, which can be shown using calculus. I wonder if the above result can be proved using mathematical induction?

I have tried but fail! Anyway, this question is also inspired by, and related to this question.

Edit:

I also found that $$\left(1+\frac{1}{n+k}\right)^n \le 3-\frac{k+1}{n}$$ for all natural number $$k$$, some large $$N$$ and $$n > N$$. This implies that $$\left(1+\frac{1}{2n}\right)^n \le 2-\frac{1}{n}.$$

And again, I can't prove any of them using Mathematical Induction.

It was shown in here that $$\left(1+\frac{1}{n}\right)^n < n$$ for $$n>3$$. I think we can be come up with a better bound, as follows:

$$\left(1+\frac{1}{n}\right)^n \le 3-\frac{1}{n}$$ for all natural number $$n$$.

The result is true for all real number $$\ge 1$$, which can be shown using calculus. I wonder if the above result can be proved using mathematical induction?

I have tried but fail! Anyway, this question is also inspired by, and related to this question.

Edit:

I also found that $$\left(1+\frac{1}{n+k}\right)^n \le 3-\frac{k+1}{n}$$ for all natural number $$k$$, some large $$N$$ and $$n > N$$. This implies that $$\left(1+\frac{1}{2n}\right)^n \le 2-\frac{1}{n}.$$

And again, I can't prove any of them using Mathematical Induction.

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# An inequality (1+1/n)^n<3-1/n using mathematical induction

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