Infinite Geometric Series

Infinite Series. If $(a_n)$ is an infinite sequence, the formal expression $a_1+a_2+\cdots$ is an infinite series, denoted $\displaystyle\sum_{k=1}^\infty a_k$. As set up in §5.6, its convergence and sum (when it exists) are governed by the limit of the partial-sum sequence $s_n=a_1+\cdots+a_n$.

The geometric series $\sum x^n$ is the prototype. Taking $s_n=x^0+x^1+\cdots+x^n=\dfrac{1-x^{n+1}}{1-x}$ ($x\ne1$): since $x^n\to0$ exactly when $|x|<1$, we get $s_n\to\dfrac1{1-x}$ under that same condition. So:

Three companion series follow by substitution:

$\sum_{n=0}^\infty(2x)^n = \frac1{1-2x},\ |x|<\tfrac12 \ \Longrightarrow\ \frac1{1-2x}=1+2x+4x^2+8x^3+\cdots$ …