Does it have any value?

Try if you can find value of below expression: \[\sqrt{1+\sqrt{\sqrt{2+\sqrt{\sqrt{\sqrt{3+\sqrt{\sqrt{\sqrt{\sqrt{4+\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5...}}}}}}}}}}}}}}}=?\]

Any solution will be appreciated

Note by Zakir Husain
1 week, 4 days ago

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1 vote

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I think it is almost equal to 1.531.53

Above expression can be written as 1+42+83+....\sqrt{1 + ^4\sqrt{2 + ^8\sqrt{3 + ....}}}

So, the rth^{th} root is 2r2^r which is increasing exponentially whereas the value inside root is increasing linearly. So, it will be equal to 11 after each root is simplified, so finally it will come as 1+1.41.53\boxed{\sqrt{1 + 1.4} \approx 1.53}. Hope it helps.

Aryan Sanghi - 1 week, 4 days ago

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even 1+2+3>2\sqrt{1+\sqrt{\sqrt{2+\sqrt{\sqrt{\sqrt{3}}}}}}>\sqrt{2}

Zakir Husain - 1 week, 4 days ago

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I know, I mean it will converge near 1.53 approx.

Aryan Sanghi - 1 week, 4 days ago

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@Aryan Sanghi Please explain your answer.

Kriti Kamal - 1 week, 4 days ago

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@Kriti Kamal Actually I'll try but I guess I can't explain better. @Zakir Husain can you please help?

Aryan Sanghi - 1 week, 4 days ago

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@Aryan Sanghi I myself don't have any idea, it came to my mind and just stuck there. That's why I have wrote this note.

Zakir Husain - 1 week, 4 days ago

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@Aryan Sanghi Please

Kriti Kamal - 1 week, 4 days ago

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@Kriti Kamal See, the rth has its root value 2r2^r, isn't it? Now, the number inside the root is only rr. So, we can see that value of root is increasing exponentially whereas the number inside root is increasing linearly. So, the root will take it closer and closer to 1 as we move right.

So, we could ignore values more than 3. So, answer is 1+42+831.53\sqrt{1 + ^4\sqrt{2 + ^8\sqrt{3}}} \approx 1.53

Hope I explained well.

Aryan Sanghi - 1 week, 4 days ago

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@Aryan Sanghi Oooo,i've understood.I thought that you have done some substitution.Nice method of approximaton..are you preaparing for jee mains???

Kriti Kamal - 1 week, 4 days ago

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@Kriti Kamal Yes, for JEE mains and advanced.

Aryan Sanghi - 1 week, 4 days ago

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@Aryan Sanghi Nice

Kriti Kamal - 1 week, 3 days ago

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Is it correct now @Zakir Husain, I have edited the solution.

Aryan Sanghi - 1 week, 4 days ago

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@Aryan Sanghi It is 1.53...\approx1.53...

Zakir Husain - 1 week, 4 days ago

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