Question

PLEASE HELP WILL FAN AND MEDAL
Does the series converge or diverge? If it converges, what is the sum? Show your work.

Answer 1

Can you help @ganeshie8

Answer 2

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Answer 3

?

Answer 4

this is a geometric series, first decide whether it converges or not

Answer 5

how do I do that?

Answer 6

Easy! there is a nice criterian for knowing when they converge/diverge
see if you can eyeball the "common ratio" from the given sum

the series converges if the common ratio is between -1 and 1

Answer 7

So it does converge. Right?

Answer 8

yes! but why ?

Answer 9

because the common ratio is in between -1 and 1

Answer 10

whats the common ratio here ?

Answer 11

-1/2?

Answer 12

Perfect! im just looking for that number because you need to use that in finding the infinite sum :)

Answer 13

Okay :)

Answer 14

so now that we know it converges, lets go ahead and find the number the series adds up to

Answer 15

look up "infinite converging geometric series formula" in your notes

Answer 16

ok

Answer 17

"An infinite geometric series converges if its common ratio r satisfies –1 < r < 1. Otherwise it diverges."

Answer 18

sorry about the question marks

Answer 19

thats right! but there will be some direct formula for the sum
see if you can find it in ur notes.. i don't seem to remember the formula hmm

Answer 20

If the common ratio r lies between –1 to 1, we can have the sum of an infinite geometric series. That is, the sum exits for .

The sum S of an infinite geometric series with –1 < r < 1 is given by the formula,

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An infinite series that has a sum is called a convergent series and the sum Sn is called the partial sum of the series.

Answer 21

Is that it?

Answer 22

Yes! thats the formula

Answer 23

you already know \(\large r = \frac{-1}{2}\)

from the sum it is easy to see that the first term, \(a_1 = -4\)

plug those in the formula and simplify

Answer 24

\[\large S = \dfrac{a_1}{1-r} = \dfrac{-4}{1- (-\frac{1}{2})} = ?\]

Answer 25

no wait thats wrong

Answer 26

ok waiting..

Answer 27

\[\frac{ 4 }{ 1.5 } = \]

Answer 28

is that right?

Answer 29

leave everything in fraction form maybe..

Answer 30

yes thats right, but fractions look and feel more better..

Answer 31

\[\frac{ 4 }{ 1 1/2 } \]

Answer 32

That is a 1 1/2

Answer 33

= \[\frac{ 8 }{3 }\]

Answer 34

\[\large S = \dfrac{a_1}{1-r} = \dfrac{-4}{1- (-\frac{1}{2})} = \dfrac{-4}{3/2} = -\dfrac{8}{3}\]

Answer 35

sorry I forgot the negative

Answer 36

So is - 8/3 the sum?

Answer 37

yes !

Answer 38

Awesome! Thank you!

Answer 39

yw:)