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Sav [38]
3 years ago
10

Find the value of

6%5D%7B2%5Csqrt%5B7%5D%7B...%7D%7D%7D%7D%7D%7D" id="TexFormula1" title="2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2\sqrt[6]{2\sqrt[7]{...}}}}}}" alt="2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2\sqrt[6]{2\sqrt[7]{...}}}}}}" align="absmiddle" class="latex-formula">
An index is going to infinity.
Mathematics
1 answer:
Leno4ka [110]3 years ago
7 0

Consider the sequence

a_1=2

a_2=2\sqrt2

a_3=2\sqrt{2\sqrt[3]{2}}

a_4=2\sqrt{2\sqrt[3]{2\sqrt[4]{2}}}

and so on. We can simplify each of these terms as

a_1=2^1

a_2=2^{1+1/2}

a_3=2^{1+1/2+1/6}

a_4=2^{1+1/2+1/6+1/24}

and so on, with the general pattern

a_n=2^{1+1/2+1/6+\cdots+1/n!}

As n\to\infty, the exponent takes on the value of e-1, since

e=\displaystyle\sum_{n\ge0}\frac1{n!}

Then the sequence a_n converges to 2^{e-1}\approx3.29044.

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