Series - The Basics — Question 10

*Question 10 For a real parameter $r$, consider $\displaystyle\sum_{n=1}^{\infty}r^{2n}$.

1. Identify the first term and common ratio as functions of $r$.

2. Determine all $r$ for which the series converges and find its sum.

3. Analyze the boundary values $r=\pm1$ and the region $|r|>1$ using the nth-term test.

*Question 10 – Solution

Expanding the first terms gives $$r^2+r^4+r^6+\cdots,$$ Thus the first term is $a=r^2$ and the common ratio is $q=r^2$. A geometric series converges precisely when $|q|<1$, so $$|r^2|<1\iff |r|<1.$$

For $|r|<1$, the geometric formula gives $$S(r)=\frac{r^2}{1-r^2}.$$ The finite partial sum provides a direct verification: $$s_N=\frac{r^2(1-r^{2N})}{1-r^2}\qquad(r^2\ne1),$$ When $|r|<1$, $r^{2N}\to0$, so $s_N\to r^2/(1-r^2)$.

At both $r=1$ and $r=-1$, every term $r^{2n}$ equals $1$, so $s_N=N\to\infty$. If $|r|>1$, then $r^{2n}$ grows rather than approaches zero; the nth-term test proves divergence. Thus the convergence set is exactly $(-1,1)$.