10.6.2 converge or diverge? alternating series

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SUMMARY

The discussion centers on the convergence of the alternating series defined by the expression $$S_n= \sum_{n=1}^{\infty} (-1)^{n+1}\frac{\sqrt{n}+6}{n+4}$$. Participants confirm that by graphing the first 10 terms, the series appears to converge towards 0. The conclusion drawn is that since the nth term approaches 0, the series converges according to the Alternating Series Test.

PREREQUISITES
  • Understanding of alternating series and their properties
  • Familiarity with the Alternating Series Test
  • Basic knowledge of limits and convergence in calculus
  • Ability to graph mathematical series
NEXT STEPS
  • Study the Alternating Series Test in detail
  • Learn about convergence tests for series, including the Ratio Test and Root Test
  • Explore graphical methods for analyzing series convergence
  • Investigate the behavior of series with varying terms, such as $$S_n= \sum_{n=1}^{\infty} \frac{1}{n^p}$$ for different values of p
USEFUL FOR

Students and educators in mathematics, particularly those focusing on calculus and series convergence, as well as anyone interested in advanced mathematical analysis.

karush
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converge or diverge
$$S_n= \sum_{n=1}^{\infty} (-1)^{n+1}\frac{\sqrt{n}+6}{n+4}$$
ok by graph the first 10 terms it looks alterations are converging to 0
 
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alternating series whose nth term $\rightarrow 0 \implies$ convergence
 

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