Sum to Infinity-

Asked Sep 10 '15 at 20:02

Active Sep 10 '15 at 20:30

Viewed 68 times

What is the sum of this infinite series $\sqrt{1+ \sqrt{2 + \sqrt{3 + \sqrt{4 + \sqrt{5 + \ldots }}}}}$?

I couldn't find the logic to solve the question. Please provide me a logic to begin with..

I don't want its full answer as I want to try it doing myself...

I have asked this question on QUORA but haven't got any answer yet..

edited Sep 10 '15 at 20:30

uranix

asked Sep 10 '15 at 20:02

shashank bhardwaj

Read this a tutorial on how to typeset mathematics on this site. – N. F. Taussig Sep 10 '15 at 20:07
Very interesting question... You could try making a recurrence relation which has terms $a_1=1, a_2=\sqrt [2] {1+\sqrt [2] {2}}$, and so on. This may not work though. – Cataline Sep 10 '15 at 20:33
Related OEIS sequence – uranix Sep 10 '15 at 20:34
2

Not really an infinite series. – zhw. Sep 10 '15 at 20:43
This seems related to Ramanujan's evaluation of:$$\sqrt{1+2\sqrt{1+3\sqrt{1+\dotsb}}}$$which equals $3$(!) – Akiva Weinberger Sep 10 '15 at 21:02
I think that's proven by showing that it's equal to the limit of $3$, $\sqrt{1+2(4)}$, $\sqrt{1+2\sqrt{1+3(5)}}$, … $\sqrt{1+\dotsb+(n-1)\sqrt{1+n(n+2)}}$, …, which is the constant sequence. – Akiva Weinberger Sep 10 '15 at 21:05

0 Answers0