Showing that $\lim_{n \to \infty}\frac {s_n} n \to 0$

Question

Let $\sum a_k$ be an infinite series and $s_n$ be the partial sums. we can assume that $\lim_{k \to \infty}ka_k = 0$. I want to show that $$\lim_{n \to \infty}\frac {s_n} n = 0$$

and am not sure how to do it. I thought about multiplying both top and bottom by $n$ so that we might use the assumption $\lim_{k \to \infty}ka_k = 0$ but this only eliminates some tail terms.

@TheSilverDoe No I haven't although now just looking it up I can see it might apply here. But is there another way to do it without it? — , Dec 13 '21 at 20:48
@AdamRubinson sorry, a typo, thikn I confused myself with another problem — , Dec 13 '21 at 20:56
Also, I'm fairly certain I have seen this question many times before on this site. — Adam Rubinson, Dec 13 '21 at 20:59
@AdamRubinson I tried searching but the results were almost all probability related — , Dec 13 '21 at 21:01

score -1 · Accepted Answer · answered Dec 13 '21 at 21:49

-1

if $\lim\limits_{k\rightarrow\infty} k*a_k=0$ then ofcourse $\lim\limits_{k\rightarrow\infty} a_k=0$. But then $\lim\limits_{n\rightarrow\infty}\frac {S_n} n=0$ by convergence of arithmetic mean.

answered Dec 13 '21 at 21:49

NivMan

147

Showing that $\lim_{n \to \infty}\frac {s_n} n \to 0$

1 Answers1