To show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$
Let $u_r, u_{r+1}$ represent the $r^{th}$ and $(r + 1)^{th}$ terms of the expansion; then
$$\frac{u_{r+1}}{u_r}=\frac{n-r+1}{r}x$$
When $r>n+1$, this ratio is negative; that is, from this point the terms are alternately positive and negative when $x$ is positive, and always of the same sign when $x$ is negative.
Now when $r$ is infinite, $Lim_{n\to \infty} \frac{u_{r+1}}{u_r} = x$ numerically ; therefore since $x < 1$ the series is convergent if all the terms are of the same sign ; and therefore a fortiori it is convergent when some of the terms are positive and some negative. $\blacklozenge$
Question 1: The Binomial expansion of $(1+x)^n$ has $(n+1)$ terms, then how can $r>n+1$?
Question 2: from this point the terms are alternately positive and negative when $x$ is positive, and always of the same sign when $x$ is negative, (How ?)