An interesting series - what does it converge to?

[kairama15]

I've lately been interested in series and how they converge to interesting values. It's always interesting to see how they end up adding up to something involving pi or e or some other unexpected solution.

I learned about the Leibniz formula back in college : pi/4 = 1/1-1/3+1/5-1/7+1/9-...
and discovered that it is simply the McLauren series for arctan(x) evaluated at 1. It's amazing that the odd integers are so involved in pi!

I've also been curious about the Basel problem involving:
pi^2/6 = 1/1^2+1/2^2+1/3^2+1/4^2+...
It's so cool that pi is still involved in the inverse squares. There is a beautiful geometric proof of this pi^2/6 result on youtube by "3Blue1Brown" called "why is pi here?". Really amazing to see how it all works out.

However, there is still one series that remains a mystery to me:
sum of 1/n^n from n=1 to n=infinity
or...
1/1^1+1/2^2+1/3^3+1/4^4+1/5^5+...
This series rapidly converges to a value according to wolfram alpha. But I was wondering if anyone knows the EXACT value rather than the approximation. And if there is a proof of it anywhere. Preferably understandable by an undergraduate math major. I'm curious - and i searched the internet for an hour trying to find something and can't find a thing!

Any help would be appreciated by you folks on this forum! :)

 

[member 587159]

Here is what you are probably after:

https://en.wikipedia.org/wiki/Sophomore's_dream

 

[dromarand]

\pi = 6 - \sum_{n=1}^{ \infty } \frac{1}{\left( n+ \frac{1}{2} \right)\left( n- \frac{1}{2} \right) } + \frac{1}{2\left( n+ \frac{1}{4} \right)\left( n- \frac{1}{4} \right) }

\phi = \sqrt{2+ \frac{1}{ \sqrt{2+ \frac{1}{ \sqrt{2+...} } } } }

 

[WWGD]

[latex] \pi = 6 - \sum_{n=1}^{ \infty } \frac{1}{\left( n+ \frac{1}{2} \right)\left( n- \frac{1}{2} \right) } + \frac{1}{2\left( n+ \frac{1}{4} \right)\left( n- \frac{1}{4} \right) } [/latex]

[latex] \phi = \sqrt{2+ \frac{1}{ \sqrt{2+ \frac{1}{ \sqrt{2+...} } } } } [/latex]

Please use $.$ to tag, i.e., please wrap your code with ##'s.

 

[dromarand]

$\pi = 6 - \sum_{n=1}^\infty \frac{1}{\left(n+\frac{1}{2}\right)\left(n-\frac{1}{2}\right)} + \frac{1}{2\left(n+\frac{1}{4}\right)\left(n-\frac{1}{4}\right)}$

$\phi = \sqrt{2+\frac{1}{\sqrt{2+\frac{1}{\sqrt{2+...}}}}}$

 

[fresh_42]

Please use $.$ to tag, i.e., please wrap your code with ##'s.

Corrected. Sometimes it is easier to hit the report button. It's less noisy.

This thread is several years old. It was never meant to sample series and their limits. Please create a new thread if you want to do so and choose a descriptive title.