Math Is Fun Forum

figo2476

Member

Registered: 2006-12-18

Posts: 1

1 + b + b^2 + ..... b^n = ?

Hi folks:
I wonder what does it equal. There is a formula, but I already forgot. Also, what is this series called in math.

Regards

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: 1 + b + b^2 + ..... b^n = ?

This is a Geometric Series. You can work out what the sum is like this:

Let x = 1 + b + b^2 + ... + b^n. [1]
Then bx = b + b^2 + b^3 + ... + b^(n+1) [2]
Subtracting [1] from [2] gives bx - x = b^(n+1) - 1.
Factorising the left side gives x(b-1) = b^(n+1) - 1.
Rearranging: x = [b^(n+1) - 1]/[b-1].

Additionally, if b is between -1 and 1, then the sum has a finite value as n goes to infinity. This is given by 1/(1-b).

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Why did the vector cross the road?
It wanted to be normal.