Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$

Question

$$S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$$ Show the convergence of $S_n$ (the method of difference more preferably)

I just began treating sequences in school, and our teacher taught that monotone increasing sequence, bounded above and monotone decreasing sequences, bounded below converge.

and so using that theorem here.. I found the $$(n+1)_{th} term$$, $$S_{n+1} = \frac1{n+1} \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n+1}\right)$$

and then subtracted the (n)th term from it

What I was able to get was... $$S_{n+1}-S_n = \frac1{(n+1)^2} - \frac{\left(1+\frac12+\dots+\frac1n\right)}{n(n+1)}.$$ ...but then this is where I get stucked, but i'm trying to prove that the sequence > 0(i.e Converges) or < 0 (i.e diverges).

Answer 1

You can check here how can this be solved:

$$\lim_{n\to\infty}\frac1n=0\implies \lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac1k=0$$

Answer 2

Hint. As an alternative to a Cesaro-like theorem, one may use the fact that $x \mapsto \dfrac1x$ is decreasing over $[1,\infty)$ to get $$ 0<1+\frac12+\frac13+\cdots+\frac1n<1+\int_1^n\frac{dx}x=1+\log n $$ giving $$ 0<\frac1n\left(1+\frac12+\frac13+\cdots+\frac1n\right)<\frac1n+\frac{\log n}n. $$

Answer 3

If you denote $H_n=\displaystyle\sum_{k=1}^n\frac1k$, one knows $H_n\sim_\infty\ln n$, hence $$S_n\sim_\infty \frac{\ln n}n\to 0.$$

A more elementary proof: \begin{align*}&S_n =\frac1n H_n>S_{n+1}=\frac1{n+1}\Bigl(H_n+\frac{1}{n+1}\Bigr)\\ \iff \enspace&(n+1)H_n>nH_n+\frac n{n+1}\iff H_n>\frac n{n+1}, \end{align*} which is true since $H_n>1>\dfrac n{n+1}$. So the sequence $(S_n)$ is decreasing and bounded from below by $0$. Apply the monotone convergence theorem.

Answer 4

Avoiding appeals to logarithms:

Writing $H_n = 1 +\frac{1}{2}+\cdots + \frac{1}{n}$, note that for $n>M$:

$$H_n=1 +\frac{1}{2}+\cdots + \frac{1}{n}=1 +\frac{1}{2}+\cdots + \frac{1}{m}+\cdots+\frac{1}{n}=H_M+\frac{1}{M+1}+\cdots+\frac{1}{n}$$

Now, each of the terms after $H_M$ is $<\frac{1}{M}$, whence $H_n<H_M+\frac{n-M}{M}$.

Thus, $\frac{H_n}{n}<\frac{H_M}{n}+\frac{n-M}{nM}=\frac{H_m-1}{n}+\frac{1}{M}$ for any positive integer $M$, as long as $n$ is larger than $M$.

From this, we deduce that $\limsup\frac{H_n}{n}\le\frac{1}{M}$ for all positive integers $M$, and thus that $\limsup\frac{H_n}{n}=0$.

As $\frac{H_n}{n}>0$, it is thus trivial to deduce that the sequence converges to $0$.