We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
164
3
avatar

Evaluate the sum \(1 + \frac{3}{3} + \frac{5}{9} + \frac{7}{27} + \frac{9}{81} + \dotsb\)

 Apr 29, 2019
 #1
avatar+176 
0

Because the sum goes on forever, we know that the sum from 5/9 on is 1, so 1 + 1 + 1 = 3

 Apr 29, 2019
 #2
avatar+23273 
+3

Evaluate the sum

\(\large{s= 1 + \dfrac{3}{3} + \dfrac{5}{9} + \dfrac{7}{27} + \dfrac{9}{81} + \dotsb }\)

 

\(\begin{array}{|rcll|} \hline s &=& 1 + \dfrac{3}{3} + \dfrac{5}{9} + \dfrac{7}{27} + \dfrac{9}{81} + \dotsb \\\\ &=& \dfrac{1}{3^0} + \dfrac{3}{3^1} + \dfrac{5}{3^2} + \dfrac{7}{3^3} + \dfrac{9}{3^4} + \dotsb + \dfrac{2n-1}{3^{n-1}} + \dotsb \\\\ \hline s_n &=& \dfrac{1}{3^0} + \dfrac{3}{3^1} + \dfrac{5}{3^2} + \dfrac{7}{3^3} + \dfrac{9}{3^4} + \dotsb + \dfrac{2n-1}{3^{n-1}} \\\\ &=& 1\cdot 3^0 + 3\cdot 3^{-1} + 5\cdot 3^{-2} + 7\cdot 3^{-3} + 9\cdot 3^{-4} + \dotsb + (2n-1)\cdot 3^{1-n} \\ \hline \end{array} \)

 

\(\begin{array}{|rcll|} \hline s_n &=& 1\cdot 3^0 + & 3\cdot 3^{-1} + 5\cdot 3^{-2} + 7\cdot 3^{-3} + 9\cdot 3^{-4} + \dotsb + (2n-1)\cdot 3^{-n+1} \\ 3^{-1} s_n &=& & 1\cdot 3^{-1} + 3\cdot 3^{-2} + 5\cdot 3^{-3} + 7\cdot 3^{-4} + \dotsb + (2n-3)\cdot 3^{-n+1}+ (2n-1)\cdot 3^{-n} \\ \hline \end{array} \\ \begin{array}{rcll} s_n-3^{-1} s_n &=& 1\cdot 3^0 +2\cdot ( 3^{-1} + 3^{-2} + 3^{-3} + 3^{-4} + \dotsb + 3^{-n+1}) - (2n-1)\cdot 3^{-n} \\ s_n(1-3^{-1}) &=& 1 +2\cdot ( \underbrace{3^{-1} + 3^{-2} + 3^{-3} + 3^{-4} + \dotsb + 3^{-n+1}}_{\text{geometric progression}} ) - (2n-1)\cdot 3^{-n} \\ \dfrac{2}{3} s_n &=& 1 +2\cdot \mathbf{ S_n } - (2n-1)\cdot 3^{-n} \\ \end{array}\\ \begin{array}{|rcll|} \hline S_n &=& 3^{-1} + & 3^{-2} + 3^{-3} + 3^{-4} + \dotsb + 3^{-n+1} \\ 3^{-1} S_n &=& & 3^{-2} + 3^{-3} + 3^{-4} + \dotsb + 3^{-n+1} + 3^{-n} \\ \hline \end{array} \\ \begin{array}{rcll} S_n-3^{-1} S_n &=& 3^{-1} - 3^{-n} \\ S_n(1-3^{-1} ) &=& 3^{-1} - 3^{-n} \\ \dfrac{2}{3} S_n &=& 3^{-1} - 3^{-n} \\ \mathbf{ S_n } &=& \mathbf{\dfrac{3}{2} ( 3^{-1} - 3^{-n} )} \\ \end{array}\)

 

\(\begin{array}{|rcll|} \hline \dfrac{2}{3} s_n &=& 1 +2\cdot \mathbf{ S_n } - (2n-1)\cdot 3^{-n} \quad | \quad \mathbf{ S_n = \dfrac{3}{2} ( 3^{-1} - 3^{-n} )} \\\\ \dfrac{2}{3} s_n &=& 1 +3\cdot ( 3^{-1} - 3^{-n} ) - (2n-1)\cdot 3^{-n} \\\\ \dfrac{2}{3} s_n &=& 1 +3^1 3^{-1} - 3^13^{-n} - (2n-1)\cdot 3^{-n} \\\\ \dfrac{2}{3} s_n &=& 1 + 1 - 3^{-n}(3 + 2n-1 ) \\\\ \dfrac{2}{3} s_n &=& 2- 3^{-n}(2n+2) \\\\ \dfrac{2}{3} s_n &=& 2- 2\cdot 3^{-n}(n+1) \\\\ \dfrac{2}{3} s_n &=& 2\Big(1- 3^{-n}(n+1)\Big) \quad | \quad : 2 \\\\ \dfrac{1}{3} s_n &=& 1- 3^{-n}(n+1) \\ s_n &=& 3\Big(1- 3^{-n}(n+1)\Big) \\ &=& 3\Big(1- \dfrac{n+1} {3^{n} }\Big) \\ &=& 3\Big(\dfrac{3^{n}-n-1} {3^{n} }\Big) \\ &=& \dfrac{3^{n}-n-1} {3^{-1}\cdot 3^{n} } \\ \mathbf{ s_n } &=& \mathbf{\dfrac{3^{n}-n-1} {3^{n-1} } } \\ \hline \end{array} \)

 

\(\begin{array}{|rcll|} \hline s = \sum \limits_{n=1}^{\infty}\dfrac{2n-1}{3^{n-1}} = \lim \limits_{n\to\infty} s_n &=& \lim \limits_{n\to\infty}\dfrac{3^{n}-n-1} {3^{n-1} } \quad | \quad \text{l'Hospital's rule} \\ &=& \lim \limits_{n\to\infty}\dfrac{n3^{n-1}-1} {(n-1)3^{n-2} } \quad | \quad \text{l'Hospital's rule} \\ &=& \lim \limits_{n\to\infty}\dfrac{n(n-1)3^{n-2}} {(n-1)(n-2)3^{n-3} } \\\\ &=& \lim \limits_{n\to\infty}\dfrac{n(n-1)3^{n-2}} {(n-1)(n-2)3^{n-2}3^{-1} } \\\\ &=& \lim \limits_{n\to\infty}\dfrac{n} { (n-2)3^{-1} } \\\\ &=& 3\cdot \lim \limits_{n\to\infty}\dfrac{n} {n-2} \\\\ &=& 3\cdot \lim \limits_{n\to\infty}\dfrac{1} {\dfrac{n-2}{n} } \\\\ &=& 3\cdot \lim \limits_{n\to\infty}\dfrac{1} {1-\dfrac{2}{n} } \quad | \quad \lim \limits_{n\to\infty} \dfrac{2}{n} = 0 \\\\ &=& 3 \cdot \dfrac{1} {1-0 } \\\\ &=& 3 \cdot \dfrac{1} {1 } \\\\ \mathbf{ s } &=& \mathbf{3} \\ \hline \end{array}\)

 

laugh

 Apr 30, 2019
 #3
avatar
0

∑[3^(1 - k) (2 k - 1) , k, 1, 1000] = 3

 Apr 30, 2019

17 Online Users

avatar