1-Compare-pi-2022e-and-e-2022-2-Compute-value-of-P-pi-

Question Number 179105 by Shrinava last updated on 24/Oct/22

1 . Compare : π^{2022 e} and e^{2022 π}

2 . Compute value of P = π^{π^{π^{\dots^{π}}}}

Answered by MJS_new last updated on 25/Oct/22

1 . same as comparing π^{e} and e^{π}

π^{e} \overset{?}{<>} e^{π}

e \ln π \overset{?}{<>} π \ln e

\frac{e}{\ln e} \overset{?}{<>} \frac{π}{\ln π}

\frac{d [\frac{x}{\ln x}]}{d x} = \frac{1}{\ln x} - \frac{1}{\ln^{2} x} {\begin{cases} < 0; 0 < x < e \\ = 0; x = e \\ > 0; x > e \end{cases}

π > e \Rightarrow \frac{e}{\ln e} < \frac{π}{\ln π} \Rightarrow e \ln π < π \ln e \Rightarrow π^{e} < e^{π} \Rightarrow

\Rightarrow π^{2022 e} < e^{2022 π}

Commented by Acem last updated on 25/Oct/22

π > e ∣_{{i t}^{'} s o k} \Rightarrow \frac{e}{\ln e} \overset{h o w ?}{<} \frac{π}{\ln π}

Commented by MJS_new last updated on 25/Oct/22

\frac{x}{\ln x} is increasing for x > e because its derivate

is > 0 as I showed

Answered by MJS_new last updated on 25/Oct/22

2 . P = + \infty

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