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Help with describing a sequence

  • #1
rock.freak667
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1. Homework Statement

Describe the following sequence

0.3,0.33,0.333,0.3333,0.33333,...


2. Homework Equations



3. The Attempt at a Solution

I tried to put the sequence as

[tex]\frac{3}{10},\frac{3}{10}+\frac{3}{100},\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}+...[/tex]

But I didn't see anything happening there. Also I would hope the answer is convergent to [itex]\frac{1}{3}[/itex] since it looks like the number of 3's tend to infinity which would be the number of 3's in [itex]\frac{1}{3}[/itex]
 

Answers and Replies

  • #2
Dick
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It's 3 times the geometric series (1/10)^n starting at n=1, isn't it? The series sums to 1/9, doesn't it?
 
  • #3
335
0
.3,.33,.333,.3333..........
> 3 [.1+.11+.111+.1111.........]
Solve in factors of 10 now ...
go ahead....
 
  • #4
tiny-tim
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Describe the following sequence

0.3,0.33,0.333,0.3333,0.33333,...
Hi rock.freak667! :smile:

You're only asked to describe the sequence.

In other words: find a formula A_n for the nth term.

For example, if the sequence is 1 8 27 64 …, then you describe it as {n³}.

Hint: As you point out, they're obviously getting closer to 1/3. So what is the difference between each term and 1/3? :smile:
 
  • #5
rock.freak667
Homework Helper
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Hi rock.freak667! :smile:

You're only asked to describe the sequence.

In other words: find a formula A_n for the nth term.

For example, if the sequence is 1 8 27 64 …, then you describe it as {n³}.

Hint: As you point out, they're obviously getting closer to 1/3. So what is the difference between each term and 1/3? :smile:
well the differences are like this

1/3-0.3=0.0333...
1/3-0.33=0.00333...
1/3-0.333=0.000333...

So the differences run in a GP of first term a=0.0333... and common ratio r=0.1

But the sum to infinity = 0.0333../0.9 = 0.333...*1/9 = 1/3 * 1/9 = 1/27..Not 1/3 as I wanted.
 
  • #6
exk
119
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rock.freak667 remember that [tex]\sum\frac{1}{10^{n}} = \frac{1}{1-\frac{1}{10}}[/tex] only when n goes from 0 to infinity. Is that the case in your example?
 
  • #7
tiny-tim
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well the differences are like this

1/3-0.3=0.0333...
1/3-0.33=0.00333...
1/3-0.333=0.000333...
Hi rock.freak667! :smile:

(remember, the question asks you to describe the sequence, and you haven't specifically done that yet.)

So 3An = 1 - what ?

And so ∑An = … ? :smile:
 

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