Prove limit with convergence tests

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SUMMARY

The discussion centers on proving the limits of two sequences using convergence tests. The first sequence, limn→∞ n*qn = 0 for |q| < 1, requires a foundational approach, while the second sequence, limn→∞ 2*n/n!, was successfully analyzed using the ratio test, yielding L = 0, confirming convergence. The user also mentions utilizing the squeeze theorem to establish the limit for the second sequence. The conversation highlights the importance of convergence tests in limit proofs.

PREREQUISITES
  • Understanding of convergence tests, specifically the ratio test.
  • Familiarity with sequences and limits in calculus.
  • Knowledge of the squeeze theorem for limit evaluation.
  • Basic algebraic manipulation skills for handling sequences.
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  • Study the application of the ratio test in greater depth, particularly for sequences involving factorials.
  • Explore the squeeze theorem and its applications in proving limits.
  • Learn about other convergence tests such as the root test and comparison test.
  • Investigate the behavior of sequences with exponential terms, particularly for |q| < 1.
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Students and educators in calculus, mathematicians focusing on sequence convergence, and anyone seeking to deepen their understanding of limit proofs using convergence tests.

esuahcdss12
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I need to prove that the limit of the sequence is as shown(0):

1.limn→∞ n*q^n=0,|q|<1
2.limn→∞ 2*n/n!
but I need to do this using the convergence tests. With the second sequence I tried the "ratio test", and I got the result

limn→∞ 2/n+1
which means that L in the ratio test is 0 and so it proves that the sequence converges, but how now should i prove that the limit is indeed 0? I can't use the L'Hopital's rule.

and for the first sequence I am not sure where to start.

can you help please?
 
Last edited:
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It looks like question 2 has a typo.
 
greg1313 said:
It looks like question 2 has a typo.

Fixed.
can anyone help with the first sequence ?
the second I manged to solve with the squeeze theory
 

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