Chapter 12 Sequences and Series - 12.4 Find Sums of Infinite Geometric Series - 12.4 Exercises - Skill Practice - Page 824: 35

$ |x| \lt 0.25$ and $S_n=\dfrac{1}{1-4x}$

Here, we have $a_n= a_1 r^{n-1}$ for the Geometric series. The sum of an infinite Geometric Series can be found using: $S_n=\dfrac{a_1}{1-r}$ First term $a_1=1$ and Common ratio $r=4x$ We know that for an infinite Geometric Sequence to converge: $|r| \lt 1 $ This gives: $|4x| \lt 1$ $\implies |x| \lt 0.25$ Thus, we have $S_n=\dfrac{1}{1-4x}$ Hence, $ |x| \lt 0.25$ and $S_n=\dfrac{1}{1-4x}$