Question

IF the series is convergent, find its sum

Answer 1

sorry, i keep losing connection

Answer 2

i cant seem to type the dam problem right! ugh

Answer 3

\[\sum_{n=1}^{\infty}[(0.7)^{n-1}-(0.4)^n]\]

Answer 4

hmm that looks the same thing we just did
we can rewrite like \[\sum (0.7)^{n-1}-\sum (0.4)^n=\sum (\frac{7}{10})^{n-1}-\sum (\frac{4}{10})^n\]

Answer 5

i didnt even think about doing that with the decimals!

Answer 6

did you check the convergency

Answer 7

well you have to do all sort of things to bring to what you want

Answer 8

no

Answer 9

they are both less than 1 and more than -1. so it is convergent, right?

Answer 10

what if it does not converge you will end up losing time for nothing

Answer 11

im not sure how to process with the n-1

Answer 12

proceed

Answer 13

seems so the limit is zero so the series converge

Answer 14

i would do \[(\frac{7}{10})^n\times \frac{10}{7}\]

Answer 15

10/7 is just constant you can throw it off the bank

Answer 16

when you multiply a fraction by its opposite, it removes the -1 from n-1?

Answer 17

well this is what i did \[(\frac{7}{10})^n (\frac{7}{10})^{-1}\]

Answer 18

and (7/10)^-1 is 10/7 as you know

Answer 19

ok, yes... so how does this affect how we add?

Answer 20

we didn't have a way of doing n-1
but we have for n
you just do we did before for each sum

Answer 21

7/10 is constant you can pull it of the first summation

Answer 22

10/7 not 7/10 sorry

Answer 23

r=7/10 and the second one r=4/10

Answer 24

i gotta go sleep working early lol :)

Answer 25

is it \[\frac{ \frac{ 7 }{ 10 } }{ \frac{ 10 }{ 10 } -\frac{ 7 }{ 10 }} - \frac{ \frac{ 4 }{ 10 } }{ \frac{ 10 }{ 10}-\frac{ 4 }{ 10 }}\]?

Answer 26

yeah but the first one don't forgot to multiply by 10/7

Answer 27

do you have the answer this time

Answer 28

so, if i throw the multiplying 10/7 in... is it \[\frac{ 7 }{ 3 }(\frac{ 10 }{ 7 })- \frac{ 4 }{ 6 }\]

Answer 29

almost lol

Answer 30

yeah that looks good

Answer 31

what is the given answer

Answer 32

dont have one this time :(

Answer 33

it is either your right or wrong

Answer 34

oh ok no problem

Answer 35

you should get 8/3

Answer 36

i got 8/3 yes, im gonna plug that in real quick

Answer 37

yep! JEEZ, you're GOOD! Thanks AGAIN!

Answer 38

I do have another question real quick tho, if you dont mind

Answer 39

http://www.wolframalpha.com/input/?i=sum+1+to+infty+%280.7%29%5E%28n-1%29
seeems 10/3 for first is good which is what we got
so the second one as well should also be good

Answer 40

earlier you said you confirmed the limit was zero and so it must be convergent. How do i do that with the n-1? that looks like infinity-1 to me

Answer 41

hmm good question
well i did exatly what i explained to you i made it 10/7 times (7/10)^n
and take the limit

Answer 42

... guess i'm forgetting my rules... is it anything raised to infinity = zero ?

Answer 43

10^n is going to infinity faster that 7^n
so it must tend to zero

Answer 44

ohh

Answer 45

oh no not anything of course

Answer 46

i see how you were looking at that....

Answer 47

because 1/infinity= zero

Answer 48

yeah it is a comparison of the two exponential functions

Answer 49

yes

Answer 50

and if the denom is going to infinity faster than numerator, it is a/infinity which= zero

Answer 51

well any k/oo=0 not just 1

Answer 52

oh yes, thats why i used an a the 2nd time

Answer 53

ok, thnks AGAIN. Really appreciate the help

Answer 54

if you were to graph that you should have
something like that

Answer 55

so it is going to zero we proceed to oo

Answer 56

very good

Answer 57

that an exponential function with base <1 and the graph has that shape

Answer 58

ohh, I see... very good visual

Answer 59

yeah you need to thing about visual too when doing such things :)

Answer 60

what about something like \[\sqrt[n]{5}\] why is this divergent?

Answer 61

do you mean \[\sum\sqrt[n]{5}\]

Answer 62

yes

Answer 63

n=1 to infty

Answer 64

hmm good question the limit is 1

Answer 65

these are the things that stump me.

Answer 66

i need a "rules list" to study or something

Answer 67

it should be zero as much as i remember

Answer 68

well i don't really have a nice answer for now
but guess it is about the nature of the radicals

Answer 69

thanks anyway :)

Answer 70

i will think about it and see :) good question
i guess i will retire for today

Answer 71

me too! have a good night

Answer 72

if i found anything interesting i will post it for you

Answer 73

good night

Answer 74

thanks! night