Define $L(k+1,k)=\frac{ \sum\_{n=0}^{k+1} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{k} {\lfloor{\pi^{n}}\rfloor} }$

${\lfloor{\pi^{0}}\rfloor}=1$,

${\lfloor{\pi^{1}}\rfloor}=3$,

${\lfloor{\pi^{2}}\rfloor}=9$,

${\lfloor{\pi^{3}}\rfloor}=31$,

${\lfloor{\pi^{4}}\rfloor}=97$,

${\lfloor{\pi^{5}}\rfloor}=306$,

${\lfloor{\pi^{6}}\rfloor}=961$,

${\lfloor{\pi^{7}}\rfloor}=3020$,

${\lfloor{\pi^{8}}\rfloor}=9488$,

${\lfloor{\pi^{9}}\rfloor}=29809$,

${\lfloor{\pi^{10}}\rfloor}=93648$

Then:

$\sum\_{n=0}^{0} {\lfloor{\pi^{n}}\rfloor}=1$,

$\sum\_{n=0}^{1} {\lfloor{\pi^{n}}\rfloor}=4$,

$\sum\_{n=0}^{2} {\lfloor{\pi^{n}}\rfloor}=13$,

$\sum\_{n=0}^{3} {\lfloor{\pi^{n}}\rfloor}=44$,

$\sum\_{n=0}^{4} {\lfloor{\pi^{n}}\rfloor}=141$,

$\sum\_{n=0}^{5} {\lfloor{\pi^{n}}\rfloor}=447$,

$\sum\_{n=0}^{6} {\lfloor{\pi^{n}}\rfloor}=1408$,

$\sum\_{n=0}^{7} {\lfloor{\pi^{n}}\rfloor}=4428$,

$\sum\_{n=0}^{8} {\lfloor{\pi^{n}}\rfloor}=13916$,

$\sum\_{n=0}^{9} {\lfloor{\pi^{n}}\rfloor}=43725$,

$\sum\_{n=0}^{10} {\lfloor{\pi^{n}}\rfloor}=137373$

Then:

$L(k+1,k)=\frac{ \sum\_{n=0}^{k+1} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{k} {\lfloor{\pi^{n}}\rfloor} }$

$L(1,0)=\frac{ \sum\_{n=0}^{1} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{0} {\lfloor{\pi^{n}}\rfloor} }=\frac{4}{1}=4$

$L(2,1)=\frac{ \sum\_{n=0}^{2} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{1} {\lfloor{\pi^{n}}\rfloor} }=\frac{13}{4}=3.25$

$L(3,2)=\frac{ \sum\_{n=0}^{3} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{2} {\lfloor{\pi^{n}}\rfloor} }=\frac{44}{13}=3.384615...$

$L(4,3)=\frac{ \sum\_{n=0}^{4} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{3} {\lfloor{\pi^{n}}\rfloor} }=\frac{141}{44}=3.20454545...$

$L(5,4)=\frac{ \sum\_{n=0}^{5} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{4} {\lfloor{\pi^{n}}\rfloor} }=\frac{447}{141}=3.170212...$

$L(6,5)=\frac{ \sum\_{n=0}^{6} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{5} {\lfloor{\pi^{n}}\rfloor} }=\frac{1408}{447}=3.149888143...$

$L(7,6)=\frac{ \sum\_{n=0}^{7} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{6} {\lfloor{\pi^{n}}\rfloor} }=\frac{4428}{1408}=3.1448863636...$

$L(8,7)=\frac{ \sum\_{n=0}^{8} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{7} {\lfloor{\pi^{n}}\rfloor} }=\frac{13916}{4428}=3.14272809...$

$L(9,8)=\frac{ \sum\_{n=0}^{9} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{8} {\lfloor{\pi^{n}}\rfloor} }=\frac{43725}{13916}=3.142066685...$

$L(10,9)=\frac{ \sum\_{n=0}^{10} {\lfloor{\pi^{n}}\rfloor} }{ \sum\_{n=0}^{9} {\lfloor{\pi^{n}}\rfloor} }=\frac{137373}{43725}=3.14174957...$

So Is the following true and how can we prove it:

$\lim\_{k\to\infty}L(k+1,k)=\pi$
?

and is it good way to approxiamte $\pi$?