Question

How do you know if two series are the same or not?

Answer 1

Is \[\sum_{n=1}^{\infty} (-\frac{ 1 }{ 2 })^{n-1}\] the same as \[\sum_{n=1}^{\infty} -(\frac{ 1 }{ 2 })^{n-1}\]

Answer 2

@phi

Answer 3

@iGreen

Answer 4

if the minus sign is inside the exponent, and the exponent is even, what happens ?
in other words, is there a difference between
-(2*2) and (-2 * -2) ?

Answer 5

Yes there is

Answer 6

But my big question, I guess, is how do I prove that they are the same or different? What proof is sufficient to say that?

Answer 7

well work out the first few terms I guess

Answer 8

I saw on a website that someone proved the a values for the two series are different, but how does that prove that the series are different?

Answer 9

well if the first few terms are different then the series must be different

Answer 10

So a series that goes -2,0,2,4,6... is different from a series that goes 0,2,4,6,8?

Answer 11

First series
first term =, 1
2nd term = -1/2

Answer 12

yes

Answer 13

Ok. I guess that is the root of my confusion. So if twos series start in different places, even if the rest of their terms are identical, the series are different?

Answer 14

oh yes

Answer 15

Ok. Thank you!!!

Answer 16

yw - the third term is different in your 2 series

1/4
and -1/4

Answer 17

yw?

Answer 18

you're welcome

Answer 19

Oh ok lol