I have a summation of b to the power of 2t. What does it tend to as t tends to infinity? b is between 0 & 1.?

Question

Hoping to make my question clearer: I know the actual expression "b to the power of 2t" will tend to 0 as t tends to infinity, but I've got a summation. So basically, what I need to know is what the sum [b^2*1 + b^2*2+.....+b^2*infinity] tends to. Thanks to all who answer!

Answer 1

Not sure if we're doing t from 0 to infinity or 1 to infinity...

Let's assume from 1

Let S = sigma(t = 1 to infinity) b^(2t)
S = b^2 + b^4 + b^6 + ...
Sb^2 = b^4 + b^6 + ....

S - Sb^2 = b^2
S(1-b^2) = b^2
S = b^2/(1-b^2)

Now if it from t=0 to infinity, then it's
S = b^0 + b^2/(1-b^2)
S = 1 + b^2/(1-b^2)
S = ((1-b^2) + b^2)/(1-b^2)
S = 1/(1-b^2)

Answer 2

It will approach zero. Numbers that are above 0 and less than 1 get really small when raised to high powers.