Expert: Paul Klarreich - 10/4/2007

Question
show the infinite series has the values S=4 at x=1/2:  S=1+2x+3x^2 =4x^3...

Answer
Questioner:   Greg
Category:  Calculus
Private:  No
 
Subject:  infinite series
Question:  show the infinite series has the values S=4 at x=1/2:  S=1+2x+3x^2 =4x^3...
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Hi, Greg,

I assume you meant to write:

S(x) = 1 + 2x + 3x^2 PLUS 4x^3 ...

If we didn't have those pesky coefficients (1,2,3,4,...) this would just be a geometric series, easily summed.  If it said:

S0(x) = 1 + x + x^2 + x^3 ...

Then we would have:

=======================================================
WARNING: USE COURIER FONT TO VIEW THIS
===================================================
        1
S0(x) = -----
      1 - x

and with  x = 1/2, that would be

         1
S0(x) = ------- = 2
       1 - 1/2

[See anecdote at the end, with a moral attached.]

So we can try the following 'split' of the series.  Let

S0 = 1 + x + x^2 + x^3 + x^4 + ...   << extra term written out
S1 =     x + x^2 + x^3 + x^4 + ...  
S2 =         x^2 + x^3 + x^4 + ...  
S3 =               x^3 + x^4 + ...  
etc.

Now I think you can see that the following is true:

S0 =      S0
S1 =    x S0
S2 =  x^2 S0
S3 =  x^3 S0      
etc.

and that:

S = S0 + S1 + S2 + S3 + ....

S = S0 + x S0 + x^2 S0 + x^3 S0 + ...

S = S0(1 + x + x^2 + x^3 + ...)

S = S0(S0) = S0^2 = 4
--------------------------------
Story:  A few years ago, my wife's doctor sent her for a stress test.  The technician instructed her to walk on a treadmill while he took readings.  After a while, my wife began to get tired and asked to stop.

Tech: Well, I need two minutes more walking; can you do that?
Wife: No, that's too much; I want to stop.
Tech: How about one more minute; you can do that, can't you?
Wife: Oh, all right.
< one minute later>
Wife: Can I stop now?
Tech: Almost done. How about 30 seconds more? You can do that, surely.
Wife: Oh, OK.
<then>
Tech: Just 15 more seconds.....
Tech: Just 7.5 more seconds.....
etc.
Well, the technician had to talk very fast at the end, but he got his two minutes of walking.

Moral: I'm not sure exactly, but the moral of this story is either:

A. You never know when you'll need advanced mathematics.
B. It sure is hard for mathematicians to get jobs these days.