Exercise 209Calculus › Single-variable calculus › Series › Find the sumHigh schoolAdded 2 Nov 2026

Geometric series with a parameter

#geometric series#convergence domain#parameters

Statement

Determine whether each series converges, and if so compute its sum:
$\sum_{k\ge0}\frac{(-1)^k}{2^k},\qquad\sum_{k\ge0}\left(\frac a{1+a}\right)^k,\qquad\sum_{k\ge0}(1-x^2)^k.$

Hint

Each is $\sum r^k$ for a ratio $r$ . It converges iff $|r|<1$ , and then the sum is $\frac1{1-r}$ .

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Solution

First. $r=-\frac12$ , so the sum is $\dfrac1{1+\frac12}=\dfrac23$ .

Second. $r=\frac a{1+a}$ (with $a\ne-1$ ). The condition $|r|<1$ means $|a|<|1+a|$ . Squaring, $a^2<1+2a+a^2$ , i.e. $a>-\frac12$ . For these $a$ the sum is $\dfrac1{1-\frac a{1+a}}=1+a$ .

Third. $r=1-x^2$ . The condition $-1<1-x^2<1$ means $0<x^2<2$ , i.e. $x\in(-\sqrt2,0)\cup(0,\sqrt2)$ . The sum is $\dfrac1{1-(1-x^2)}=\dfrac1{x^2}$ .

Remark. The sum is a function of the parameter, and it blows up as $r\to1^-$ , which is exactly the edge of the convergence domain.

Geometric series with a parameter

Statement

Solution

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