Question

Use an area argument to show that ln2<1\ln 2<1.

Solution

Verified
Answered 2 years ago
Answered 2 years ago

Consider the interval [1,2][1,2] then for all t[1,2]t\in[1,2] we have 1t1\leq t which implies 1t1t[1,2]\dfrac{1}{t}\leq 1\quad \forall t\in[1,2].

Then using Sub dominance rule we get

121tdt<121dt    ln2<t12ln2<(21)ln2<1\begin{aligned}\int_{1}^{2}\dfrac{1}{t}\:dt&< \int_{1}^{2}1\:dt\\ \implies \ln{2}&< t|_{1}^{2}\\ \ln{2}&< (2-1)\\ \ln{2}&< 1\end{aligned}

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