Learn English – How to say “±” in English

• The absolute value of the difference between S and
• the sum from i equals one through i equals n of
• the function f evaluated at t sub i times the width of each i
• is less than epsilon.

If it is clear that i and n are one-indexed, then "the sum from i equals one through i equals n" can be replaced by "the sum of the first n terms". "The width of each i" is an interpretation of "delta i".

• The function is one divided by the quantity x plus one close quantity, all divided by the square root of x.
• The integral from zero to infinity of the function d x
• equals the limit as s goes to zero of the integral from s to 1 of the function d x
• plus the limit as t goes to infinity of the integral from 1 to t of the function d x.

"Goes to" can be replaced by "goes toward", or (as Damkeng suggests) "tends to", or (as J.R. suggests) "approaches".

I often use a notation like "integral from x equals a to x equals b" instead of "integral from a to b". I also often say "to positive infinity" instead of "to infinity".

I "factored out" the definition of the function in the first sentence. If the function were easy to say (such as "x squared"), I would not "factor out" the definition of the function. Instead, I would include the "x squared" in the statements of the integrals, a la Damkeng's answer.

Formally it is

Ten to the power of nine

but more commonly:

Ten to the nine or Ten to the ninth

The use of Base 9 has a very different meaning.