Series - The Basics — Question 3

Question 3 – Solution

Step 1: Derive the finite identity.

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 $$s_N=\frac{1-x^{N+1}}{1-x}.$$

Step 2: Determine the convergent parameter region.

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

Step 3: Check the excluded cases.

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.