+0  
 
+1
222
2
avatar
Find the sum of the infinite series

(-3/7)^2+(-3/7)^3+(-3/7)^4+(-3/7)^5+...
Guest Feb 6, 2012

Best Answer 

 #2
avatar+26329 
+8

No, this sum has a finite limit:

$$\sum_{k=2}^{\inf}(\frac{-3}{7})^k=\frac{9}{70}$$

See the following figure;

sum2 

 

 

 In this respect it is more like the infinite series 1 + 1/2 + 1/4 + 1/8 etc. which has the finite sum of 2. In other words it is a geometric series, the sum to infinity of which is given by:

sum = a0/(1-r) where a0 is the first term [(-3/7)2 here] and r [= (-3/7)] is the ratio between successive terms.  Because r is less than 1, successive terms get smaller, r→0, and the sum converges to a finite value.

$${\mathtt{sum}} = {\frac{{{\mathtt{\,-\,}}\left({\frac{{\mathtt{3}}}{{\mathtt{7}}}}\right)}^{{\mathtt{2}}}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}\left({\frac{{\mathtt{3}}}{{\mathtt{7}}}}\right)\right)}} \Rightarrow {\mathtt{sum}} = {\mathtt{0.128\: \!571\: \!428\: \!571\: \!428\: \!6}}$$

$${\frac{{\mathtt{9}}}{{\mathtt{70}}}} = {\mathtt{0.128\: \!571\: \!428\: \!571\: \!428\: \!6}}$$

Just noticed this is a very old unanswered question!!

Alan  May 18, 2014
Sort: 

2+0 Answers

 #1
avatar+77 
0

infinity since it is infinite, just like 1+2+4+8+16...

oskar20106  May 18, 2014
 #2
avatar+26329 
+8
Best Answer

No, this sum has a finite limit:

$$\sum_{k=2}^{\inf}(\frac{-3}{7})^k=\frac{9}{70}$$

See the following figure;

sum2 

 

 

 In this respect it is more like the infinite series 1 + 1/2 + 1/4 + 1/8 etc. which has the finite sum of 2. In other words it is a geometric series, the sum to infinity of which is given by:

sum = a0/(1-r) where a0 is the first term [(-3/7)2 here] and r [= (-3/7)] is the ratio between successive terms.  Because r is less than 1, successive terms get smaller, r→0, and the sum converges to a finite value.

$${\mathtt{sum}} = {\frac{{{\mathtt{\,-\,}}\left({\frac{{\mathtt{3}}}{{\mathtt{7}}}}\right)}^{{\mathtt{2}}}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}\left({\frac{{\mathtt{3}}}{{\mathtt{7}}}}\right)\right)}} \Rightarrow {\mathtt{sum}} = {\mathtt{0.128\: \!571\: \!428\: \!571\: \!428\: \!6}}$$

$${\frac{{\mathtt{9}}}{{\mathtt{70}}}} = {\mathtt{0.128\: \!571\: \!428\: \!571\: \!428\: \!6}}$$

Just noticed this is a very old unanswered question!!

Alan  May 18, 2014

13 Online Users

avatar
avatar
avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details