Question

Can you help me solve an infinite geometric sequence

Answer 1

A geometric series like

$\displaystyle \sum_{n=1}^\infty \dfrac{1}{\alpha^n}$

Converges if and only if $|\alpha|>1$. If this is the case, the sum equals

$\displaystyle \sum_{n=1}^\infty \dfrac{1}{\alpha^n} = \dfrac{1}{\alpha-1}$

So, in your case, you have convergence if and only if

$|2+a|>1 \iff 2+a>1 \lor 2+a-1 \lor a$

And if this is the case, the sum equals

$\displaystyle \sum_{n=1}^\infty \dfrac{1}{(2+a)^n} = \dfrac{1}{2+a-1} = \dfrac{1}{a+1}$