Series - The Basics — Question 3

*Question 3 For a real parameter $x$, consider $\displaystyle\sum_{n=0}^{\infty}x^n$.

1. For $x\ne1$, derive the finite identity $s_N=(1-x^{N+1})/(1-x)$.

2. Determine every $x$ for which the infinite series converges and find its sum there.

3. Analyze $x=1$, $x=-1$, and $|x|>1$ separately.

*Question 3 – Solution

Let $s_N=1+x+\cdots+x^N$. Then $xs_N=x+x^2+\cdots+x^{N+1}$. Subtracting the second equation from the first gives $$(1-x)s_N=1-x^{N+1},$$ so for $x\ne1$ division by $1-x$ yields $$s_N=\frac{1-x^{N+1}}{1-x}.$$

If $|x|<1$, then $x^{N+1}\to0$. Therefore the partial sums converge and $$\sum_{n=0}^{\infty}x^n=\frac1{1-x}.$$

At $x=1$, $s_N=N+1\to\infty$. At $x=-1$, the partial sums alternate between $1$ and $0$ and have no limit. If $|x|>1$, the terms $x^n$ do not approach zero, so the nth-term test proves divergence. Thus convergence occurs exactly for $|x|<1$; checking both endpoints separately is essential.