Hi and welcome to the Matladpi blog! In this post I will concentrate on a particular math problem.

I love sequences and especially series. That is why I will sometimes make a post about these topics. Like, right now.

Let there be a series S = x + x^2 + x^3 + ... > 0 that does converge. How would you prove that this implies 0 < x < 1? Think about it and then proceed by reading past the line.

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Solution:

S = x + x^2 + x^3 + ...

S = x(1 + x + x^2 + x^3 + ...)

S = x(1 + S)

x(1 + S) = S

x = S/(1 + S)

S < 1 + S

S/(1 + S) < 1

x < 1 (upper bound)

S > 0

1 + S > 1 + 0 = 1

S/(1 + S) > 0/1 = 0

x > 0 (lower bound)

Thus 0 < x < 1. Njäf! said.

I am Jesse Sakari Hyttinen and I will see you in the next post!