Question # Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. sum_{n=1}^infty(3x)^n

Series
ANSWERED Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x.
$$\sum_{n=1}^\infty(3x)^n$$ 2021-02-26

Consider the series
$$\sum_{n=1}^\infty(3x)^n$$
Consider a G.P series $$\sum_{n=1}^\infty r^n$$
Series is convergent for $$|r|<1$$
Sum of this infinite series is given by
$$S_\infty=\frac{a}{1-r}$$
$$\sum_{n=1}^\infty(3x)^n=(3x)^1+(3x)^2+(3x)^3+(3x)^4....$$
$$\sum_{n=1}^\infty(3x)^n=(3x)+(3x)^2+(3x)^3+(3x)^4....$$
The given series is a Geometric progression with the
first term $$=3x$$
common ratio $$=3x$$
The given series is convergent for
$$|3x|<1$$
$$\Rightarrow-1<3x<1$$
$$\Rightarrow-\frac13$$
Sum of the series $$\sum_{n=1}^\infty(3x)^n$$ for $$-\frac13$$
Here
$$a=3x,r=3x$$
$$S_\infty=\frac{a}{1-r}$$
$$S_\infty=\frac{3x}{1-3x}$$
Series is convergent for $$-\frac13$$
Sum of the series is $$\frac{3x}{1-3x}$$