# An interesting series - what does it converge to?

• A
• kairama15
In summary, the conversation discusses the topic of series and their convergence to interesting values, particularly those involving pi and e. The Leibniz formula and the Basel problem are mentioned as examples, with the latter having a proof on YouTube. The conversation also mentions a series that remains a mystery and requests for help in finding its exact value and proof.
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! :)

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

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

Last edited by a moderator:
dromarand said:
$\pi = 6 - \sum_{n=1}^{ \infty } \frac{1}{\left( n+ \frac{1}{2} \right)\left( n- \frac{1}{2} \right) } + \frac{1}{2\left( n+ \frac{1}{4} \right)\left( n- \frac{1}{4} \right) }$

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

##\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
WWGD said:
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.

## 1. What is convergence in a series?

Convergence in a series refers to the property where the terms of the series approach a specific value as the number of terms in the series approaches infinity. If the terms of a series do not approach a specific value, the series is said to diverge.

## 2. How do you determine if a series converges?

To determine if a series converges, you can use various tests such as the ratio test, comparison test, or integral test. These tests help determine if the terms of the series approach a finite value as the number of terms increases.

## 3. What does it mean for a series to diverge?

If a series does not converge, it is said to diverge. This means that the terms of the series do not approach a specific value as the number of terms in the series increases, and the sum of the series does not exist.

## 4. Can a divergent series have a sum?

No, a divergent series cannot have a sum. If a series does not converge, it means that the terms do not approach a specific value, and therefore, the sum of the series does not exist.

## 5. What is the importance of determining convergence in a series?

Determining convergence in a series is crucial in mathematics as it helps in understanding the behavior of the series and whether the sum of the series exists. Convergent series have a finite sum, while divergent series do not, making convergence an essential concept in mathematical analysis.

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