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Homework Help: Convergent or Divergent

  1. Jul 3, 2014 #1
    1. The problem statement, all variables and given/known data
    Which of the series, diverge or converge ∑ 5^n/(4^n +3 )

    2. Relevant equations

    3. The attempt at a solution Taking the limit as n→∞ we have (5^n ln 5)/ (4^n ln 4) , my question is here how does it become like this, which part am I missing here?
  2. jcsd
  3. Jul 3, 2014 #2
    Oh I see, they have used L'hopital rule since we got infinity over infinity
  4. Jul 3, 2014 #3


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    A necessary condition for [itex]\sum a_n[/itex] to converge is [itex]\lim_{n \to \infty} a_n = 0[/itex] (it is not a sufficient condition; the series [itex]\sum n^{-1}[/itex] diverges). Here [tex]
    \lim_{n \to \infty} \frac{5^n}{4^n + 3}
    [/tex] is calculated using L'hopital's rule. Since the limit is not zero the sum does not converge.
  5. Jul 3, 2014 #4


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    If ##\Sigma a_n## converges, then ##\lim(a_n) = 0##.

    If ##\lim(a_n) ≠ 0##, then ##\Sigma a_n## diverges.

    Alternatively, you could apply the comparison test + the geometric test with ##|r| ≥ 1##.
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