Question

Convergent or divergent
16-8+4-...

Answer 1

do you know what kinda series that is?

Answer 2

hint:
is there a common ratio?

Answer 3

if so what is it?

Answer 4

No, I have to figure out the infinite geometric series

Answer 5

there is a common ration since it is a geometric series

Answer 6

the geometric series here has a ratio of (-1/2)

Answer 7

do you see that now?

Answer 8

has a common ratio of (-1/2)*

Answer 9

Yes

Answer 10

So is it divergent?

Answer 11

now since you just need to determine if it is divergent or convergent
you just need to tell me
if you have |r|<1
---
if |r|<1 then it is convergent
if |r|>=1 then it is divergent

Answer 12

you have r=-1/2
so which inequality is true ?

Answer 13

oh, its convergent. <

Answer 14

yep yep

Answer 15

thank you! can you answer another?

Answer 16

okay

Answer 17

Find the sum of each infinite series, if it exists
440+220+110+...

Answer 18

let's see
what kinda series is this?

Answer 19

can you tell?

Answer 20

have you looked for a common ratio?

Answer 21

i think its an infinite one

Answer 22

if there is a common ratio, it is a geometric series

Answer 23

i looked for one but really couldnt figure it out

Answer 24

220/440=?
110/220=?

Answer 25

okay so 1/2

Answer 26

yes both of them are equal to 1/2
therefore there is a common ratio of 1/2
therefore since it has a common ratio it is a geometric series

Answer 27

use the same test as we did before

Answer 28

you mean the convergent and divergent thing?

Answer 29

if so its convergent since its less than 1

Answer 30

the geometric convergeny test:
if |r|<1 then it is convergent
if |r|>=1 then it is divergent

Answer 31

and yes

Answer 32

if we had
2,4,8,...
this would be a different story

Answer 33

the common ratio here would be 2

Answer 34

so 2+4+8+... would diverge

Answer 35

and divergent since its over 1?

Answer 36

right

Answer 37

okay yeah i get it.! so if i were to find the sum of the last series? how would i do that?

Answer 38

if we have -2,4,-8,...
then the common ratio is -2
and so -2+4+(-8)+...
is divergent since |-2|=2>1

Answer 39

the formula is this:
\[\sum_{n=0}^{\infty}a r^n=\frac{a}{1-r}\]

Answer 40

so you have a=440
and r=1/2

Answer 41

enter those in

Answer 42

hmmm cant seem to find it on my calculator. D:

Answer 43

find what/

Answer 44

the formula to plug the numbers into

Answer 45

\[\frac{440}{1-\frac{1}{2}} =?\]
a division sign
and subtraction sign?

Answer 46

(440) div (1-1 div 2)

Answer 47

so you think your calculator has the formula in it?

Answer 48

what kinda calculator do you have?

Answer 49

TI 84 Plus by Texas Instruments

Answer 50

it looks like you can use the calculator to by pass using the formula

Answer 51

but you have to know the put it in sigma notation

Answer 52

440,220,110,...
first number is 440
440,440(1/2),440(1/2)^2,...
this can be written as
\[\sum_{k=0}^{\infty}440 \cdot (\frac{1}{2})^k\]
so it says you can enter seq(440(1/2)^k, 0,inf)
it looks like

Answer 53

then you can use the sum command

Answer 54

sum(Ans)

Answer 55

that seems like a lot of work

Answer 56

I think using the a/(1-r) is pretty easy

Answer 57

okay yeah that way seemed a bit easier

Answer 58

so does that big greek looking E have any significance?

Answer 59

is that what you are confused about?

Answer 60

that big sigma thing means we are adding a whole bunch of terms

Answer 61

example:
\[\sum_{k=0}^{5}k^2 \text{ means } 0^2+1^2+2^2+3^2+4^2+5^2\]

Answer 62

the sigma sign tells us we are adding
the k=0 underneath means we going to starting adding terms starting with k=0
the 5 on top means that is where we will stop

Answer 63

well i missed class today. Then learned a new section and when i got my work after school she gave me a 40 problem packet and told me it was due tomorrow. so im just now trying to learn what all these words and symbols mean.

Answer 64

so instead of writting the series in sigma notation I could have wrote:
\[a+ar+ar^2+ar^3+ar^4 + \cdots =\frac{a}{1-r}\]

Answer 65

where |r|<1

Answer 66

so to do these it has to be convergent.?

Answer 67

yes you can only use that formula if you have |r|<1

Answer 68

ok sounds good

Answer 69

ok so when i plug in the numbers i got 880. is that the final answer?

Answer 70

well if youre still there. you were a big help thank you and sorry im a dumbass

Answer 71

880 is correct

Answer 72

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