Evaluating Series Homework Statement

[Ocasta]

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.<br /> r = \frac{11}{24}<br />

<br /> \sum _{i=1} ^\inf nr^n<br />

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

Homework Equations

<br /> \sum _{i=1} ^\inf nr^n \rightarrow <br /> n \sum _{i=1} ^\inf r^n \rightarrow<br /> n \frac{1}{1-r}<br />

The Attempt at a Solution

<br /> \sum _{i=1} ^\inf nr^n \rightarrow <br /> n \sum _{i=1} ^\inf r^n \rightarrow<br /> n \frac{1}{1-r}<br />

<br /> \frac{1}{1-r}<br />
This is a number greater than one,
<br /> \frac{24}{13}<br />

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

 

[Ray Vickson]

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.


<br /> r = \frac{11}{24}<br />

<br /> \sum _{i=1} ^\inf nr^n<br />

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

Homework Equations

<br /> \sum _{i=1} ^\inf nr^n \rightarrow <br /> n \sum _{i=1} ^\inf r^n \rightarrow<br /> n \frac{1}{1-r}<br />

The Attempt at a Solution

<br /> \sum _{i=1} ^\inf nr^n \rightarrow <br /> n \sum _{i=1} ^\inf r^n \rightarrow<br /> n \frac{1}{1-r}<br />

<br /> \frac{1}{1-r}<br />
This is a number greater than one,
<br /> \frac{24}{13}<br />

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