Question

How would I go about finding the value of this infinite sum? I just want to know the strategy, I have the answer.

Answer 1

\[\sum_{n=0}^{\infty} \frac{n}{10^n} \]

Answer 2

humm
1/10+2/100+3/1000+.....=0.1234567910111213141516.......

Answer 3

hehe :P

Answer 4

how much ur gonna type :O

Answer 5

Yeah, well it's precisely 10/81. I derived it myself, but I really worked backwards and can't see how I'd work it without just luckily finding it the wrong way.

Here's what I did:

\[1^2=. \bar 9 ^2=(.9+.09+.009+...)^2=.9(.9+.09+.009+...)+.09(.9+.09+.009+...)+...\]

Noticing that there will only be one .81 term, two .081 terms, etc... \[1*.81+2*.081+3*.0081+...\]

Yeah you can see how I got there now, yeah?

Answer 6

mmm just try to expand the sumation thats is the tric

Answer 7

But I want to go backwards without knowing this weird way. I don't really see how expanding the summation really works.

Answer 8

it is work :)
1/10 + 2/100 +3/1000+4/10000+...=0.1+0.02+0.003+0.0004+...

Answer 9

ill show u another method

Answer 10

Yeah but your way doesn't get the exact answer, 10/81 which is really what I want, not a decimal approximation.

Answer 11

if u wanna something more neat

Answer 12

\(\large \sum \limits_{n=0}^{\infty}x^n = \dfrac{1}{1-x}\)

differentiate both sides and plugin x = 1/10

Answer 13

\(1^2=. \bar 9 ^2\\=(.9+.09+.009+...)^2 \\ =.9(.9+.09+.009+...) +.09(.9+.09+.009+...)+...\)

Answer 14

Yeah thanks I should have tried that thanks @ganeshie8

Answer 15

Thanks for being helpful @BSwan I appreciate it.

Answer 16

hey @Kainui how did u jump from \(1*.81+2*.081+3*.0081+...\) to \(\dfrac{10}{81}\) ? im still struggling to see the link lol

Answer 17

Oh well from there that's really just all 1, so I made it into the summation: \[1=\sum_{n=1}^{\infty} \frac{n*.81}{10^{n-1}}\] combining the constant parts .81 and 10 turns it into:
\[1=8.1\sum_{n=1}^{\infty} \frac{n}{10^n}\]
so I pulled it out and got 1/8.1 which is the same as 10/81.

Answer 18

so by chance u knew its 81 ??
or did u tried another method and get that??

Answer 19

There was no chance involved, uhh... I don't really know what part is unclear but I can try to explain anything that seems like I made a guess or something. I stayed up all night so I might have accidentally left out important details, so tell me so I can help you guys out in understanding what's going on. @_@

Answer 20

Let:

x= 1/10 + 2/100 + 3/1000 + ...

10x = 1 + 2/10 + 3/100 + ...

Subtracting ,

9x = 1 + 1/10 + 1/100 + 1/1000 + ... = 10/9

Hence:

x = 10/81

Why to complicate things when it can be done simply

Answer 21

@No.name that works too I guess it just didn't seem very obvious to me that we could do that either. Nifty tricks all around. =P

Answer 22

brilliant ! thats the euler approach :P
i think i got Kainui's method... its not complicated - its just a simple matter of collecting terms in a product : (a+b+c+...)(x+y+z+...)

Answer 23

Euler again , how nice is it :P

Answer 24

I don't know euler approach but that's common sense

Answer 25

its gauss i guess lol^

Answer 26

http://mathcentral.uregina.ca/QQ/database/QQ.02.06/jo1.html

Answer 27

Now you've got me thinking I can work the approach backwards to get even messier ones to behave like a geometric series 2 steps away rather than 1 step away. Now I want to generalize it to n steps away and play around lol.

Answer 28

he was 9 yrs old i guess :P

Answer 29

who

Answer 30

gauss

Answer 31

what r u talking abt gauss 9 year old?

Answer 32

-.- nvm

Answer 33

lol me neither, i meant gauss would have approached this problem like the way @No.name solved it :)

Answer 34

What is Bswan talking about ?

Answer 35

i said i cant see gauess or eluer in this lol
when ganeshie said about gauss and provide the link i said he was 9 thats all ^^

Answer 36

She is saying, Gauss was 9 years old when he figured out the formula for sum of first 100 numbers

Answer 37

oh lol okay

Answer 38

I found an interesting series that you can solve because of this: \[\sum_{n=1}^{\infty} \frac{n^2}{10^n}\]

Kinda interesting what kinds of things you can play around with, anyone know any other interesting series?