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I am a non-mathematician who knows some elemententary calculus ans I want to prove that the sequence $(x_n)$ given by

$$ x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big) $$

is decreasing. Is there an elegant way to show this?

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welcome,compute $x_{n+1}$ and find the difference $x_{n+1}-x_{n}$ if the difference is positive then its increasing,if negetive then its decreasing –  Jonas12 Mar 4 at 20:29
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@Jonas12 I don't think that is an easy calculation to see. –  Calvin Lin Mar 4 at 20:39

4 Answers 4

This does not answer your question but might be helpful. Let $x>0$ be a real number. Then it follows from $$\log(x) = \int_1^x \frac{dt}{t}$$ and the estimates $$2-t <\frac{1}{t}<1-\frac{(t-1)x}{x+1}$$ for $t\in(1,1+1/x)$ that

$$ x-\frac{1}{2} < x^2\log\left(1+\frac{1}{x}\right)<x-\frac{1}{2}+\frac{1}{2(x+1)}. $$

In particular $$\lim_{x\to\infty}x^2\log\left(1+\frac{1}{x}\right)-x=-\frac{1}{2}$$ which might or might not be what you are after.

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You want to show that $x_n$ is monotonically decreasing, or in other words that $\frac{d}{dn} x_n$ is always non-positive. The derivative is not very friendly looking, but you get $$ \frac{d}{dn} x_n = \frac{-2 \sqrt{n} + 2(n + \sqrt{n}) log(\frac{1}{\sqrt{n}}+1)-1}{2(n+\sqrt{n})} $$ You can plug in some values for $n$ to convince yourself that this is $always$ negative for positive $n$. More rigorously, look at the numerator. Certainly the first term will give you a negative and so will the last one. The middle term will give you a positive, but approaches zero and will never be greater in magnitude than the other two terms.

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The simplest way might be to expand (n+1)^(1/2) and (n+1)^(-1/2) in the expression for x_(n+1) using the binomial theorem then show that x_(n+1) - x_n is positive.

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  1. as @user130512 said, you could prove that $x_{n+1}$ is greater than $x_n$. Sometimes, it can help to look at the ratio between the two, that is $x_{n+1}/x_n$. If it is greater than $0$, $x_{n+1}$ is greater than $x_n$.

  2. I think you also should be able to look at the derivative of $x_n$ to deduce whether it is positive or negative. So, positve $ \frac{d(x_n)}{dn}$ means it is increasing while negative would mean it is decreasing. Note that it can be increasing for one period but decreasing for another, so, as in your case, you would need to prove that $ \frac{d(x_n)}{dn}$ is smaller than $0$ for all values of $n$ to show that it is negative since the derivative will not be a constant.

The derivative in your case is:

$$\dfrac{2(n+\sqrt{n})\log\left(\tfrac1{\sqrt{n}}+1\right)+1}{2(n+\sqrt{n})}$$

As you can see, we only need to find what values would make this function equal to zero. Because of this, we can work with the nominator only, $$(n+\sqrt{n})\times \log\left( \frac{1}{\sqrt{n}} +1\right)+1 =0$$

You can also see it graphically in the derivative:

enter image description here

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