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Formulas for sequences

  • Thread starter soothsayer
  • Start date
  • #1
422
3

Homework Statement


For the following series, write formulas for the sequences an,
Sn and Rn, and find the limits of the sequences as n-->infinity
latex2png.2.php?z=100&eq=1-\frac{1}{2}%2B\frac{1}{4}-\frac{1}{8}%2B\frac{1}{16}-....jpg



Homework Equations


N/A


The Attempt at a Solution


an is easy, =
latex2png.2.php?z=100&eq=\frac{%28-1%29^n%20}{2^n%20}.jpg

the limit of which does not exist.

This is where I get stuck, I know Sn=
latex2png.2.php?z=100&eq=\sum_{k%3D1}^{n}\frac{%28-1%29^k}{2^k}.jpg

But I don't know how to come up with an actual formula for the sum. Furthermore, I'm not sure I even know what Rn is, a Reimann sum? How do I go about doing that? I can't find anything in my notes.
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
Maybe it helps if you write it like
[tex]S_n = \sum_{k = 1}^n \left( - \frac{1}{2} \right)^k[/tex]
(hint: |1/-2| < 1, geometric series).

We cannot smell what Rn is supposed to mean either. Perhaps a recursive formula? Or a Riemann sum? Or a rest term (i.e. defined by [itex]\sum_{k = 0}^\infty a_k = S_n + R_n[/itex]) ?
 
Last edited:
  • #3
422
3
Ooh, thanks for the tip! That does make things a little clearer, I'll try that out. As for what Rn stands for, the problem in the book didn't elaborate, and I don't have the book with me to reference.
 
  • #4
CompuChip
Science Advisor
Homework Helper
4,302
47
If you really want to do this assignment now, I suggest trying the first and last option that I gave.
The recursive formula is quite trivial and will probably take you about 5 second to write down.
The formula for the rest is another nice exercise manipulating geometric series and fractions :-)
 

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