Totally three expressions which may help in calculating partial sums

Now I have discovered two more expressions which can help in partial sums.

So all in all I have three expressions which help in calculating partial sums. So many infinite series are lot closer. And so is Jesus.

Intuitions and methods , both are now available to me. I feel good. 

Have a nice day !!

I have used the above partial sums formula to compute 1/2 from 1/3 terms. And also e.

I don't think our education system teaches beyond infinite Geometric series and Taylor series for infinite series.  

I feel very deeply that infinite series should be taught to class 11 and class 12 and BSc courses.  Partial sums methods should be used. Infinite series are the most predominant human thinking.  And not knowledge gaining. Because knowledge can be wrong.  But a convergent infinite series is not. Though I realize I am mixing philosophy with mathematics but I mean good. And am partially correct. 

By changing the plus operator inside the braces to multiplication symbol and restricting to k terms, we can get m to the power k. Not sigma from 1 to k of m to the power k when we have the plus operator. This is the Reimann Zeta functions for positive integers. 

A calculation towards the Basel problem 

And a calculation towards Apery's constant.

Similarly,  by changing the plus operator to multiplication operator in sigma 1 to k of m factorial we get only m factorial.