You need to find a balance between the two.

Help me evaluate the infinite series of

$\displaystyle 1 + \sqrt {2 + \sqrt[3] {3+ \sqrt[4] {4 + \sqrt[5] {5 +......}}}}$ .

#Calculus

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Please enter the answer correct to 3 decimal places.

Bryan Lee Shi Yang - 6 years, 2 months ago

And also give full step solutions (as much as possible) , bcs. I haven't figured out yet!

Bryan Lee Shi Yang - 6 years, 2 months ago

Wait....you changed the problem -_- I was working on the previous one :V

Krishna Sharma - 6 years, 2 months ago

Haha XD

Bryan Lee Shi Yang - 6 years, 2 months ago

I don't have any solution but out of curiosity, just to find out, I approximated it and got 2.911.

I made an equation ((((x-1)^2-2)^3-3)^4-4)^4........=x which gives 2.911

Archit Boobna - 6 years, 2 months ago

I think that the answer is about 2.911

Archit Boobna - 6 years, 2 months ago

Ramanujan maybe?

Krishna Ar - 6 years, 2 months ago

The only thing I know is that it is less than 3, because of an identity by Ramanujan.Since it is above 2.9, we can find a good approximation.I dont think a closed form exists.

Bogdan Simeonov - 6 years, 2 months ago

please solve and post a nice solution to this

Ritam Baidya - 6 years, 2 months ago

Awesome question!

John Young - 6 years, 2 months ago

>>>import math
>>>ans = 0.0
>>>for x in xrange(10, 0, -1):
>>>    ans = math.pow((x + ans), 1.0/x)
>>>print("%.50f" % (ans))
2.91163921624582400227154721505939960479736328125000

Using $12$ or above to starts iteration gives equal amount precision, about $50$ digits.

Damiann Mangan - 6 years, 2 months ago

Same method, but did it manually.

Archit Boobna - 6 years, 2 months ago

Then does your number have a symbol, I mean Greek symbol or anything?

Bryan Lee Shi Yang - 6 years, 2 months ago

Thanks for all your suggestions. I will post it into a question.

Bryan Lee Shi Yang - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$