Is the following statement is true?
I need to determine whether the following statement is true or false.
$\lim_{n\to \infty}\left( \dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{n+n}\right)=\lim_{n\to \infty}\dfrac{1}{n+1}+\lim_{n\to \infty}\dfrac{1}{n+2}+...+\lim_{n\to \infty}\dfrac{1}{n+n}=0+0+...+0=0 .$
Anyone can help me? Thanks.
$\endgroup$ 32 Answers
$\begingroup$Hint:Use a Riemman Integral $$\lim_{n\to \infty}\left( \dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{n+n}\right)=\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n\frac{1}{1+\frac{k}{n}}=\int_0^1\frac{1}{1+x}dx$$
$\endgroup$ $\begingroup$It is not true.
Observe that for each $n\in \mathbb{N}$,
$$\dfrac{1}{n+1}+\dfrac{1}{n+2}\dots \dfrac{1}{n+n}\ge\underbrace{\dfrac{1}{n+n}+\dfrac{1}{n+n}\dots \dfrac{1}{n+n}}_{\text{n terms}}=\dfrac{n}{n+n}=\dfrac{1}{2}.$$
Therefore $\lim\limits_{n\to \infty}\left( \dfrac{1}{n+1}+\dfrac{1}{n+2}\dots \dfrac{1}{n+n}\right)\ge \lim\limits_{n\to \infty}\dfrac{1}{2}=\dfrac{1}{2} \ne0.$
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