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Evaluating a Series

  • Thread starter Ocasta
  • Start date
  • #1
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Homework Statement


First, I'd like to thank everybody a head of time. You guys have been an enormous help.

Second, I don't mind telling you that I'm finding sequences and series extremely frustrating. I usually pick this stuff up like nobody's business.

My problem is attached, but I will copy it down as I understand it.


[itex]
r = \frac{11}{24}
[/itex]

[itex]
\sum _{i=1} ^\inf nr^n
[/itex]

Mysteriously, this can be rewritten as
[itex]
\sum_{i=1} ^n ir^i = \frac{ nr^{n+2} - (n+1)r^{n+1} + r }{ (1 - r)^2 }
[/itex]


Homework Equations


[itex]
\sum _{i=1} ^\inf nr^n \rightarrow
n \sum _{i=1} ^\inf r^n \rightarrow
n \frac{1}{1-r}
[/itex]


The Attempt at a Solution



[itex]
\sum _{i=1} ^\inf nr^n \rightarrow
n \sum _{i=1} ^\inf r^n \rightarrow
n \frac{1}{1-r}
[/itex]

[itex]
\frac{1}{1-r}
[/itex]
This is a number greater than one,
[itex]
\frac{24}{13}
[/itex]

So as n goes to infinity, the number just gets bigger and bigger right? Diverges to infinite is, apparently, not the answer.
 

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  • #2
Ray Vickson
Science Advisor
Homework Helper
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Homework Statement


First, I'd like to thank everybody a head of time. You guys have been an enormous help.

Second, I don't mind telling you that I'm finding sequences and series extremely frustrating. I usually pick this stuff up like nobody's business.

My problem is attached, but I will copy it down as I understand it.


[itex]
r = \frac{11}{24}
[/itex]

[itex]
\sum _{i=1} ^\inf nr^n
[/itex]

Mysteriously, this can be rewritten as
[itex]
\sum_{i=1} ^n ir^i = \frac{ nr^{n+2} - (n+1)r^{n+1} + r }{ (1 - r)^2 }
[/itex]


Homework Equations


[itex]
\sum _{i=1} ^\inf nr^n \rightarrow
n \sum _{i=1} ^\inf r^n \rightarrow
n \frac{1}{1-r}
[/itex]


The Attempt at a Solution



[itex]
\sum _{i=1} ^\inf nr^n \rightarrow
n \sum _{i=1} ^\inf r^n \rightarrow
n \frac{1}{1-r}
[/itex]

[itex]
\frac{1}{1-r}
[/itex]
This is a number greater than one,
[itex]
\frac{24}{13}
[/itex]

So as n goes to infinity, the number just gets bigger and bigger right? Diverges to infinite is, apparently, not the answer.
You switched summation indices incorrectly. The series is either sum_{i} i*r^i or sum_{n} n*r^n; it is NOT sum_{i} n*r^n or whatever. Go back and read the question more carefully.
Remember: |r| = 11/24 is less than 1---that matters a lot.

RGV
 

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