Question

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.

Answer 1

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

Answer 2

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

Answer 3

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.

Answer 4

I got A?

Answer 5

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

Answer 6

yes:)

Answer 7

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

Answer 8

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

Answer 9

***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
\[ \frac{1-r^\infty}{1-r} \]

Answer 10

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

Answer 11

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

Answer 12

Yes, I agree

Answer 13

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

Answer 14

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.

Answer 15

Ok, so we can cross offf C and D?

Answer 16

Then the answer would be A? Right?

Answer 17

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

Answer 18

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

Answer 19

Thanks so much<3