Twenty Million Chances to Get it Wrong

You friend believes she's found a very efficient algorithm for generating \( e \) (the base of the natural logarithm). However, she wants to check that it's correct.

She asks you to find the digit sum of the digit sum of the first twenty million digits after the decimal mark in \( e \) to ensure that her computed value matches the correct calculated value.

What is answer she should hope to have?

Details and assumptions

• The digit sum of a number is the sum of all its digits. For example the digit sum of 9123 is \(9 + 1 + 2 + 3 = 15\).
• The "digit sum of the digit sum" of 9123 is the digit sum of 15, which would be 6.
• One possible definition of \( e \) is \( \displaystyle \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n \) .