## anonymous one year ago Identify whether the series summation of 12 open parentheses 3 over 5 close parentheses to the i minus 1 power from 1 to infinity is a convergent or divergent geometric series and find the sum, if possible.

1. anonymous

$\sum_{i=1}^{\infty} 12\frac{ 3}{ 5}^i-1$

2. anonymous

It's suppose to be 12(3/5)^i-1

3. anonymous

This is a convergent geometric series. The sum cannot be found. This is a convergent geometric series. The sum is 30. This is a divergent geometric series. The sum cannot be found. This is a divergent geometric series. The sum is 30.

4. anonymous

I got A?

5. phi

$\sum_{i=1}^{\infty} 12\left(\frac{ 3}{ 5}\right)^{i-1}$?

6. anonymous

yes:)

7. phi

I would change the i=1 to i=0 (and i-1 in the exponent to i) just so it looks "more standard" $\sum_{i=0}^{\infty} 12\left(\frac{ 3}{ 5}\right)^{i} \\ 12\sum_{i=0}^{\infty} \left(\frac{ 3}{ 5}\right)^{i}$ there is a standard formula for this

8. anonymous

no. It's (3/5)^i-1 not i=1

9. phi

***no. It's (3/5)^i-1 not i=1*** yes, but I did this: let i= j+1 then i-1 becomes (j+1-1) = j and in the summation, the lower limit i=1 becomes j+1= 1 or j=0 and the upper limit stays infinity then I renamed j to i the summation part is $\frac{1-r^\infty}{1-r}$

10. anonymous

Ohhh!!! Ok, I get it now! I realise that's what you were doing

11. phi

I did that so it looks like the "rule" $\sum_{i=0}^N r^i = \frac{1-r^N}{1-r}$

12. anonymous

Yes, I agree

13. phi

r is 3/5 and we know r*r*r*... gets smaller and smaller, and if we do it enough times, it approaches zero. thus the summation is $\frac{1-0}{1-\frac{3}{5}} = \frac{1}{\frac{2}{5}}= \frac{5}{2}$ multiply by the leading 12, to get 12*5/2 = 30

14. phi

divergent means the sum "diverges" (fancy word for "moves away" ... from a fixed number) in other words, you don't get an answer if it diverges. cross off any answer with "diverges" because we found an answer.

15. anonymous

Ok, so we can cross offf C and D?

16. anonymous

Then the answer would be A? Right?

17. phi

choice A says there is no answer. But we can find the answer. (see up above)

18. anonymous

Sorry, I meant B :) Since we got 30 as an answer

19. anonymous

Thanks so much<3