How to get the sum (geometric ∞ sequence)?

Question

How to sum this?

∞

Σ (1/2)^r

r=0

What does the r mean? Isn't it usually n?

Completely lost, please help, thank you!

Answer 1

The sigma indicates a summation of terms each of which is identified by the following formula (1 / 2)^r.

The limits r = 0 to infinity indicate the range of integer values assigned to r.

The formula is therefore formal shorthand for:

(1 / 2)^0 + (1 / 2)^1 + (1 / 2)^2 + (1 / 2)^3 + ... to infinity

= 1 + 1 / 2 + 1 / 4 + 1 / 8 + ... to infinity.

If the upper limit were n instead of infinity, the series would be:

(1 / 2)^0 + (1 / 2)^1 + (1 / 2)^2 + (1 / 2)^3 + ... + (1 / 2)^(n - 1) + (1 / 2)^n

= 1 + 1 / 2 + 1 / 4 + 1 / 8 + ... + 1 / 2^(n - 1) + 1 / 2^n.

It is therefore irrelevant what letter is used in the formula and in the limit, as long as they match.

This series is geometric, as you say in the question title. Each term is the previous one multiplied by 1 / 2.

The sum to n terms is:

(1 / 2)[ 1 - (1 / 2)^n ] / [ 1 - (1 / 2) ]

Letting n tend to infinity gives:

(1 / 2) [ 1 - 0 ] / [ 1 - (1 / 2) ]

= (1 / 2) / (1 / 2)

= 1.