Finding the Sum of a Series: Proving Solutions

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SUMMARY

The discussion revolves around finding the sum of the infinite geometric series defined by the formula \(\sum_{n=2}^\infty \frac{1}{4^n}\). The user initially struggles with proving the convergence of the series but realizes that it is indeed a geometric series with a common ratio of \(\frac{1}{4}\). The sum converges to \(\frac{1/4}{1 - 1/4} = \frac{1/4}{3/4} = \frac{1}{3}\). This conclusion is reached by recognizing the properties of geometric series.

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  • Understanding of geometric series and their convergence
  • Familiarity with the formula for the sum of an infinite geometric series
  • Basic knowledge of limits and series in calculus
  • Ability to manipulate algebraic expressions
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  • Study the properties of geometric series in detail
  • Learn about convergence tests for infinite series
  • Explore the derivation of the sum formula for geometric series
  • Practice solving various types of series problems, including non-geometric series
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Homework Statement


I'm having trouble finding the sum of a series. I am able to perform the task for "geometric" series, where there is an "a" or an "r" value. But take the following problem for instance:
[tex] \sum_{n=2}^\infty \frac{1}{4^n}[/tex]
I'm just not sure how to approach this problem and "prove" a solution. I know that since the numbers being added are getting smaller and smaller this sum likely converges, and I suspect it converges at (1/4). But again, how do I show this?

Homework Equations





The Attempt at a Solution

 
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That is a geometric series. It's (1/4)^n.
 
Ah, indeed it is. I don't know how I missed that, I guess this post can be deleted.
 

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