지수곡선 삼각형 넓이 극한

$P _{n} ( n , 2 ^{n} )$, $Q _{n} ( n , 3 ^{n} )$, $P _{n-1} ( n - 1 , 2 ^{n-1} )$ $\displaystyle S _{n} = P _{n} Q _{n} P _{n-1} = \frac{1}{2} ( 3 ^{n} - 2 ^{n} )$ $\displaystyle T _{n} = \sum\limits _{k = 1} ^{n} S _{k} = \sum\limits _{k = 1} ^{n} \frac{1}{2} ( 3 ^{k} - 2 ^{k} )$ $\displaystyle = \sum\limits _{k = 1} ^{n} \frac{3 ^{k}}{2} - \sum\limits _{k = 1} ^{n} \frac{2 ^{k}}{2}$ $\displaystyle = \frac{1}{2} \frac{3 ( 3 ^{n} - 1 )}{3 - 1} - \frac{1}{2} \frac{2 ( 2 ^{n} - 1 )}{2 - 1}$ $\displaystyle = \frac{3 ^{n+1}}{4} - 2 ^{n} + \frac{1}{4}$ $\displaystyle \lim\limits _{n \rightarrow \infty} {\frac{T _{n}}{3 ^{n}}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {\frac{\frac{3 ^{n+1}}{4} - 2 ^{n} + \frac{1}{4}}{3 ^{n}}}$$\displaystyle = \frac{3}{4}$