Formulas for Sequences: Finding Limits and Sums for an, Sn, and Rn

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Homework Help Overview

The discussion revolves around finding formulas for sequences an, Sn, and Rn related to a series, as well as determining the limits of these sequences as n approaches infinity. The subject area includes sequences, series, and potentially Riemann sums.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to define the sequences and their limits but expresses uncertainty about deriving a formula for Sn and the meaning of Rn. Some participants suggest writing Sn in a specific form and hint at its relation to geometric series. Others question the definition of Rn, proposing it could refer to a recursive formula or a Riemann sum.

Discussion Status

Participants are actively exploring different interpretations of the problem, with some offering guidance on how to approach the formulation of Sn. There is acknowledgment of the ambiguity surrounding Rn, and suggestions are made to clarify its meaning.

Contextual Notes

The original poster lacks access to the textbook for reference, which may limit their understanding of Rn and the context of the problem. There is also a mention of homework constraints that may affect the discussion.

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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.
 
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Maybe it helps if you write it like
S_n = \sum_{k = 1}^n \left( - \frac{1}{2} \right)^k
(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 \sum_{k = 0}^\infty a_k = S_n + R_n) ?
 
Last edited:
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.
 
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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