Question

Determine whether each infinite geometric series converges or diverges. If the series converges, state the sum.
-10 - 20 - 40 - ...

Can someone help me with this? Preferably step by step directions?
I'll give a medal to whoever helps me!!

Answer 1

-(10+20+40+...)

are the terms getting smaller? or bigger?

Answer 2

a sum only converges if the terms get smaller and smaller, fast enough. a convergent series needs to be adding terms to it that are essentially equal to 0 in order to stymie their growth.

Answer 3

@amistre64 I'm not sure if it's adding or not. I checked on Mathway and it said it had no solution but the sum was S3 = -70 so I guess it's subtracting?

Answer 4

a sequence (a list) of numbers can be generated in a few ways: geometrically is one way to generate a list of the numbers.

adding up the numbers in the list is what this question is about, given that the numbers were generated geometrically.

Answer 5

ANY list of numbers can only converge to an actual amount if and only if the list of numbers get closer and closer to zero, fast enough.

Answer 6

you can tell by the way the numbers are presented that they are not getting anywhere close to zero as they move on ...

Answer 7

since they are not getting anywhere close to zero, they are simply making the sum bigger and bigger and bigger, so there is no value that they settle in on ... it grows to -infinity

Answer 8

@amistre64 Which means what? It diverges?

Answer 9

i would say it doesnt converge .. which is the same thing as saying it diverges in most cases.

Answer 10

if the sum of the terms of a sequence do not settle down and conform themselves to some static value .... then they do not converge.

Answer 11

@amistre64 Okay, so it diverges but still has a sum?

Answer 12

it has no sum if it diverges ...

Answer 13

if it doesnt settle in on some value , then there is no value we can attribute to it.

Answer 14

imagine the straw the breaks the camels back ... its a saying

1 straw is light
2 straws is light
3 straws .... the weight just adds on and on

eventually the camel caves under the pressure of the weight of the straws since the sum of the weights do not even out to some known value ... it just grows and grows

Answer 15

spose we know we can hold at most, 50 lbs

if what we are adding to the load is steadily decreasing in weight such that the sum total never exceeds 50 lbs, then we can hold an infinite number of those items. their combined weight will eventually limit to some value.

Answer 16

@amistre64
Right, I knew that. So the answer is the geometric series has no sum, it diverges. Thanks!

Answer 17

*since the geometric

Answer 18

correct :)

Answer 19

in general a geometric sum will only converge if its common ratio is between -1 and 1

Answer 20

in this case .... the common ratio is 2, so there is no chance of convergence