Solve Infinite Series: Sum of -(5/4)^n

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SUMMARY

The infinite series \(-\sum (\frac{5}{4})^n\) diverges due to the common ratio \(r = -\frac{5}{4}\), which has an absolute value greater than 1. According to the convergence criteria for geometric series, specifically \(\sum (ar)^n = \frac{a}{1-r}\) when \(0 < |r| < 1\), this series does not meet the necessary condition for convergence. The test for divergence confirms that the series approaches infinity, thus establishing its divergence definitively.

PREREQUISITES
  • Understanding of geometric series and their convergence criteria
  • Familiarity with the concept of series divergence
  • Knowledge of mathematical notation and summation
  • Basic algebra skills for manipulating series terms
NEXT STEPS
  • Study the convergence tests for series, focusing on the Ratio Test
  • Explore the properties of geometric series in greater detail
  • Learn about series convergence in the context of real analysis
  • Investigate examples of divergent series and their implications
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Students studying calculus, particularly those focusing on series and sequences, as well as educators teaching mathematical convergence concepts.

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[SOLVED] Infinite series help

Homework Statement



[tex]\sum- (\frac{5}{4})^n[/tex]
i=infinity and n=0

Homework Equations


Convergence of a geometric series
[tex]\sum (ar)^n = a/(1-r) when 0<|r|<1[/tex]

The Attempt at a Solution


I have to explain why this series diverges or converges. The test for divergence gives an answer of infinity so it diverges. The terms are 1, -5/4, 25/16, -125/64, 625/256... To me it looks like a geometric series with r=|-5/4| which diverges because |-5/4|[tex]\geq[/tex] 1. Is this correct?
 
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Correct.
 
thanks!
 

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