MATH

# Is raised to the ln of any number we get that number

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is raised to the ln of any number, we get  that number. For example,  e ln 12  would equal 12 or  e ln 3 x 4  would  equal 3x-4. However, we already know that the limit of  n 1 n  gives us an

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indeterminate form. We can use the power rule to get  ln n 1 n  ( ln x a →a∙ ln x ¿ to a form that would allow us to find the limit. The  following steps will demonstrate this process. i. lim n→∞ e ln n 1 n   ii. Power rule for  raised to  ln n 1 n n 1 n = ¿ 1 n ln n ln ¿  =  ln n n iii. lim n→∞ e ln n n     e lim n→ ∞ ln n n   iv. lim n→∞ ln n n   v. Apply l’Hopital’s Rule to  ln n n lim n→∞ 1 n 1     lim n→∞ 1 n    0 vi. lim n→∞ e ln n 1 n  becomes  e 0 =1 5) Revisit limit in Step 3:  1 e lim n→∞ 100 1 n ∙n 1 n  =  1 e 1 1 1 e B. Determine the convergence or divergence of the given infinite series according to the  results of the root test.  The infinite series,  n = 1 100 n e n , converges absolutely by the root test because when  ρ < 1 , the series converges absolutely. The   for the infinite series was equal to  1 e C. Hass, J., & Weir, M. D. (2008).   Thomas calculus: Early transcendentals . Boston:  Pearson Addison-Wesley.
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• Summer '17
• lim, Natural logarithm, n N

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