Problem 5. Consider the series >an, where an(In(ln(n))-In n.n=1(a) Show, by taking logarithms, that an = n¬In(ln(ln n)).(b) Show that In(ln(ln n)) > 2 if nece?(c) Show that this series converges.

Answered: Problem 5. Consider the series > an,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem 5. Consider the series >
an, where an
(In(ln(n))-In n.
n=1
(a) Show, by taking logarithms, that an = n¬In(ln(ln n)).
(b) Show that In(ln(ln n)) > 2 if n
ece?
(c) Show that this series converges.

Expert Solution

Step 1

Given: $\sum_{n = 1}^{\infty}$ $a_{n}$ , $a_{n} = (\ln (\ln {(n))}^{- \ln n}$

$(a)$ Taking $\log_{n}$ both sides

$\log_{n} a_{n} = \log_{n} (\ln (\ln {(n))}^{- \ln n} \log_{n} a_{n} = - \ln (n) \log_{n} (\ln (\ln (n))) (∵ \log m^{n} = n \log m) \log_{n} a_{n} = - \ln (n) \times \frac{\log_{e} (\ln (\ln (n)))}{\log_{e} n} (∵ \log_{a} b = \frac{\log_{e} b}{\log_{e} a}) \log_{n} a_{n} = - \ln \times \frac{\ln (\ln (\ln (n)))}{\ln} \log_{n} a_{n} = - \ln (\ln (\ln (n))) \Rightarrow a_{n} = n^{- \ln (\ln (\ln (n)))} (u \sin g \log_{a} x = y \Rightarrow x = a^{y})$

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