In calculus, the sum of an infinite geometric series whose first term is 1/20 is given by the complex fraction

1/20
_____
1-r where r is the common ratio between the terms. Simplify this expression.

(1/20)/(1-r)

= 1/(20 - 20r) or 1/(20(1-r))

To simplify the complex fraction, we'll start by finding the common ratio, denoted as r. In an infinite geometric series, each term is obtained by multiplying the previous term by the common ratio r.

Given that the first term is 1/20, the second term will be (1/20) * r, the third term will be (1/20) * r^2, and so on.

Now, we know that the sum of an infinite geometric series is given by the formula:

S = a / (1 - r),

where S represents the sum, a is the first term, and r is the common ratio.

We are given the first term a = 1/20, so we can substitute it into the formula:

S = (1/20) / (1 - r).

To simplify the expression, let's work on the denominator first. We multiply the numerator and the denominator by the conjugate of the denominator, which is (1 - r) itself.

Simplifying the denominator:

(1 - r) * (1 - r) = 1 - r - r + r^2,

Combining like terms:

1 - 2r + r^2.

Now, we can rewrite the expression as:

S = (1/20) / (1 - 2r + r^2).

The expression is now simplified.