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avatar+68 

Compute the sum
\(1 + \frac{1}{5} + \frac{2}{5} + \frac{2}{25} + \frac{4}{25} + \frac{4}{125} + \frac{8}{125} + \frac{8}{625} + \dots\)

 Apr 7, 2024

Best Answer 

 #1
avatar+214 
+1

This sum if really a combination of 2 geometric series.

First geometric series: \(1\over{5}\),\(2\over{25}\),\(4\over{125}\), and so on

Second geometric series: \(2\over{5}\), \(4\over{25}\), \(8\over{125}\), and so on

use: \(a_1\over1-r\) formula: The sum of the first geometric series would be: \(1\over{5}\)\(\div\)\(3\over5\)=1/3

2nd would be: 2/5 \(\div\) 3/5 = 2/3

1/3+2/3 = 1

We still didn't add the first term: 1

So 1+1 = 2

smileycoolsmiley

 #1
avatar+214 
+1
Best Answer

This sum if really a combination of 2 geometric series.

First geometric series: \(1\over{5}\),\(2\over{25}\),\(4\over{125}\), and so on

Second geometric series: \(2\over{5}\), \(4\over{25}\), \(8\over{125}\), and so on

use: \(a_1\over1-r\) formula: The sum of the first geometric series would be: \(1\over{5}\)\(\div\)\(3\over5\)=1/3

2nd would be: 2/5 \(\div\) 3/5 = 2/3

1/3+2/3 = 1

We still didn't add the first term: 1

So 1+1 = 2

smileycoolsmiley

 #2
avatar+128794 
0

Nice, AD  !!!!

 

 

cool cool cool

 Apr 7, 2024

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