Value of an infinite series

Question

How can we find the value of $(1+x)(1-x+x^2-x^3+x^4....... \textrm{infinity})$. I think it will be 1 but not too sure of it. I think all the terms will get cancelled only 1 remains but how to show it?

Answer 1

How can we find the value of $(1+x)(1-x+x^2-x^3+x^4- \dots)$ ?

This expression doesn't have a "value" per se. But this notation is usually interpreted as $\lim_{n \to \infty} (1+x)(1-x+x^2- \dots + (-x)^n)$ and value of this limit, whenever it exists, is considered to be a value of your expression (otherwise we say that expression doesn't have value at all).

I think it will be 1 but not too sure of it. I think all the terms will get cancelled only 1 remains but how to show it?

For the interpretation above, we get $\lim_{n \to \infty}(1+x)(1-x+x^2- \dots + (-x)^n) = \lim_{n \to \infty}(1-(-x)^{n+1})$. This limit exists (in traditional sense) iff $x \in [-1,1)$ and equals $0$ for $x=-1$, $1$ for other values of $x$. Note how "all the terms" never get cancelled.

Also, one can say that for $x < -1$ the limit equals $-\infty$, this statement has a number of possible interpretations. Be careful, though, since using $\pm \infty$ as a "value" can be tricky. Personally, I prefer to say that the limit simply doesn't exist in this case.