limit with summation

[man111]

For a given sequence $a_1,a_2,\ldots,a_n$ if $\lim\limits_{n\to \infty}a_n=a$, Then $\lim\limits_{n\to \infty}\dfrac{1}{\ln(n)}\sum\limits_{r=1}^{n}\dfrac{a_r}{r}$ is

[vezni]

Директно можем да ползваме теоремата на Щолц, понеже [tex]\ln(n)[/tex] е строго растяща и неограничена.
[tex]\lim_{n\to\infty}\frac{\sum_{r=1}^{n+1}\frac{a_r}{r}-\sum_{r=1}^{n}\frac{a_r}{r}}{\ln(n+1)-\ln(n)}=\lim_{n\to\infty}\frac{\frac{a_{n+1}}{n+1}}{\ln\frac{n+1}{n}}=[/tex]

[tex]=\lim_{n\to\infty}\left(a_{n+1}\cdot\frac{\frac 1n}{\ln\left(1+\frac 1n\right)}\cdot\frac{n}{n+1}\right)=a\cdot 1\cdot 1=a[/tex]

[tex]\Rightarrow \lim_{n\to\infty}\frac{1}{\ln n}\sum_{r=1}^{n}\frac{a_r}{r}=a[/tex] от теоремата на Щолц.