Testing convergence of sequence

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SUMMARY

The discussion focuses on the convergence of two series: the series sum[sin(0.4n) / nπ]² converges due to its dominance by the convergent p-series 1/n². In contrast, the series sum[sin(0.4n) / nπ] diverges, as indicated by the divergence of the 1/|n| series. The p-series test was applied to the first series, while the root test was utilized for the second series, confirming the conclusions drawn about their convergence behaviors.

PREREQUISITES
  • Understanding of series convergence tests, specifically the p-series test and root test.
  • Familiarity with trigonometric functions and their properties, particularly sine functions.
  • Knowledge of mathematical notation and series summation.
  • Basic calculus concepts, including limits and infinite series.
NEXT STEPS
  • Study the application of the p-series test in greater detail.
  • Learn about the root test and its conditions for convergence.
  • Explore the properties of trigonometric series and their convergence behaviors.
  • Investigate other convergence tests, such as the ratio test and comparison test.
USEFUL FOR

Mathematicians, students studying calculus, and anyone interested in series convergence analysis will benefit from this discussion.

jlu
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how can i can the convergence of sum[sin (0.4)n / n pi]2 and that sum[sin (0.4)n / n pi] diverges, n is between -infinity and + infinity
 
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jlu said:
how can i can the convergence of sum[sin (0.4)n / n pi]2 and that sum[sin (0.4)n / n pi] diverges, n is between -infinity and + infinity

The first series converges because it is dominated by 1/n^2 series, which is convergent. The second is harder - the 1/|n| series diverges, but this is only a clue.
 
i used p-series test for the first one and root test for the second one. i stand to be corrected if i used the wrong rules
 

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