Apumerghata (NEW)

Apumerghata is an operation. It comes from the Gujarati words "અપૂર્ણાંક (apūrṇāṅka)", "ઉમેરો (umērō)", and "ઘાત (ghāta)", meaning "fraction", "add", and "exponent", respectively.

Symbol
Its symbol is four dots.

Formula
The formula for apumerghata is n = $${\frac{1}{1}^\sqrt{1}+\frac{1}{2}^\sqrt{2}+\frac{1}{3}^\sqrt{3}+...+\frac{1}{n}^\sqrt{n}}$$.

For numbers with a decimal point, the formula is n = $$\frac{1}{1}^\sqrt{1}+\frac{1}{2}^\sqrt{2}+\frac{1}{3}^\sqrt{3}+...-(\frac{1}{n}^\sqrt{n}-\frac{1}{(n+\lceil{n})\div2}^\sqrt{(n+\lceil{n})\div2}-\frac{1}{\lceil{n}}^\sqrt{\frac{1}{\lceil{n}}})$$.

Using this formula, π would be equal to approximately 1.5456410313, and e would be equal to approximately 1.5013201067.

With this new formula, you can now use apumerghata for any number, regardless of whether or not it has a decimal point. Therefore, it is better than the original apumerghata.

Numbers from 1-20 using apumerghata
Note: 1 is the only rational number in this sequence. Every number afterwards is irrational. As you can see, it will never reach 2. It has its own constant, the Apumerghata Constant, which is equal to ∞. You can read the first 100 digits by clicking on the link.