At what x does this series converge

Question

At what $x$ does the series $\sum_{0}^{\infty} (1-(\frac{\sqrt{x}}{3}))^n $ and prove for those $x$ it converges to $f(x)=\frac{3}{2+ \sqrt{x}}$ .

I know the general term must tend to zero in order to have a chance for convergence. So for $x=9$, I get zero. How do I continue?

Answer 1

It is a geometric series of the form $$\sum a^n $$ which, you should know, converges $$\iff |a|<1$$ the sum is then $\frac {1}{1-a} $.

so, your series converges $$\iff |1-\frac {\sqrt {x}}{3}|<1$$ $$\iff -1 <1-\frac {\sqrt {x}}{3}<1$$ $$\iff 0 <\frac {\sqrt {x}}{3}<2$$ $$\iff 0<\sqrt{x}<6$$ $$\iff 0 <x <36$$

Answer 2

Hint:

A geometric series $\sum_{n=0}^{\infty}a^{n}$ converges to $\frac{1}{1-a}$ for $|a|<1$.