Warning: include_once(/var/www/nurkholis/data/www/nurkholis.web.id/wp-content/plugins/wp-super-cache/wp-cache-phase1.php): failed to open stream: No such file or directory in /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/advanced-cache.php on line 22

Warning: include_once(): Failed opening '/var/www/nurkholis/data/www/nurkholis.web.id/wp-content/plugins/wp-super-cache/wp-cache-phase1.php' for inclusion (include_path='.:/opt/php74') in /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/advanced-cache.php on line 22

Warning: Cannot modify header information - headers already sent by (output started at /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/advanced-cache.php:22) in /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/plugins/wp-editormd/src/Main.php on line 113

Warning: include(/var/www/nurkholis/data/www/nurkholis.web.id/wp-content/plugins/wp-super-cache/wp-cache-base.php): failed to open stream: No such file or directory in /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/plugins/wp-super-cache/wp-cache.php on line 115

Warning: include(): Failed opening '/var/www/nurkholis/data/www/nurkholis.web.id/wp-content/plugins/wp-super-cache/wp-cache-base.php' for inclusion (include_path='.:/opt/php74') in /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/plugins/wp-super-cache/wp-cache.php on line 115

Warning: include_once(/var/www/nurkholis/data/www/nurkholis.web.id/wp-content/plugins/wp-super-cache/ossdl-cdn.php): failed to open stream: No such file or directory in /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/plugins/wp-super-cache/wp-cache.php on line 138

Warning: include_once(): Failed opening '/var/www/nurkholis/data/www/nurkholis.web.id/wp-content/plugins/wp-super-cache/ossdl-cdn.php' for inclusion (include_path='.:/opt/php74') in /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/plugins/wp-super-cache/wp-cache.php on line 138

Warning: Cannot modify header information - headers already sent by (output started at /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/advanced-cache.php:22) in /var/www/nurkholis/data/www/nurkholis.web.id/wp/wp-content/plugins/wp-super-cache/wp-cache-phase2.php on line 1539
Problem - Jumlah Barisan Repunit | Catatan Nurkholis

Problem – Jumlah Barisan Repunit

Problem:

Jika diketahui $B_n$ merupakan bilangan yang terdiri dari angka 1 sebanyak $n$. Misal $B_3=111$, $B_7=1111111$. Tentukan nilai dari
$$lim_{ntoinfty}  sum_{k=1}^n frac{B_k}{10^n}$$


Solusi:
Perhatikan bahwa $displaystyle B_k=111…111 = frac{10^k-1}{9}$
sehingga:
$$
begin{aligned}
lim_{ntoinfty}  sum_{k=1}^n frac{B_k}{10^n} & = lim_{ntoinfty}  sum_{k=1}^n frac{10^k-1}{9cdot10^n}\
&= lim_{ntoinfty}  frac{1}{9}left(sum_{k=1}^n frac{10^k}{10^n}-sum_{k=1}^n frac{1}{10^n}right)\
&= lim_{ntoinfty}  frac{1}{9}left(sum_{k=1}^n frac{1}{10^{n-k}}- frac{n}{10^n}right)\
&= lim_{ntoinfty}  frac{1}{9}left(sum_{m=0}^n frac{1}{10^{m}}- frac{n}{10^n}right)\
&= lim_{ntoinfty}  frac{1}{9}left(frac{1left(1-frac{1}{10^n}right)}{1-frac{1}{10}}- frac{n}{10^n}right)\
&=frac{1}{9}cdotfrac{10}{9}-lim_{ntoinfty}frac{n}{9cdot10^n}\
&=frac{10}{81}
end{aligned}
$$

Tinggalkan komentar

%d blogger menyukai ini: