#PythaShastri more expressions on e and π

The above are some more expressions on e and π.
e is an important constant, which occurs in nature. Euler worked a lot and did a lot on e.
Another expression of π is also pasted. This expression reaches to π quite efficiently and can be used in some graduation and post-graduation courses. Any help needed on inclusion in curricula will be obliged.
How do I make expressions? There are certain numbers and theories in me. There is a pattern in these. These patterns, I try to explain in mathematical expressions.  As an example (-1) ^ (n+1) is actually talking of odds and evens. For when n is odd + sign is returned. And when n is even - sign is returned. So (-1)^(n+1) talks of odds and evens.

Meaning of my life

The life of an individual is bound by two things. One is his own principles. The system prevalent in his society. By society, I mean the place he lives;the friends and relatives; and maybe other factors.
Earlier, I tried to lead an accountable life. And I thought accountable life is parmo dharma. But people around did not seem to care for accounting.
So, I changed the meaning of my life. It is no longer accounting. But conscience and quality are the two words driving my life.
About society, I hate helplessness. And I expect society to follow successful models of working than to be a society that sets examples. I feel India should follow China or US rather than spending energy on making India great.

These days . . .

It was impossible to go ahead further on mathematics series. I know and can explain the most famous constants of e, pi, gamma and phi. But my pi expressions are not the fastest. I wanted to read books on higher number theories. But these are tough. Therefore, I have stopped mathematics research.
Cartooning and caricatures are possible only to a certain extent. Linear progress as a function of linear time is not possible. Therefore, I have restricted myself in cartoons and caricatures too.
I am working on chess and now my elo rating is 1520. Here is the snapshots.

Working and improving on chess is more fruitful.
1520 is good for clubs. But not possibly good enough to force GMs in chess to teach me. 1800 they say is the requirement.
So, I shall try for 1800 . . .