How do I go about proving that $$\mathbf\sum _{n=1}^{\infty }\:\left(\frac{1}{2^n}+\frac{1}{3^n}\right)$$ converges?
Using the ratio test: $$\frac{|a_n+1|}{|a_n|}<1$$ I was able to simplify the problem to this: $$\mathbf \lim _{n\to \infty \:}\left(\frac{2^{-n-1}+3^{-n-1}}{2^{-n}+3^{-n}}\right)$$ which gave me $\frac{1}{2}$. However, my textbook says the series converges to $\frac{3}{2}$. I retried the calculation but cannot see where I went wrong. Could anyone please give a hint? Also, how do I prove the convergence without finding the limit?
Yes, your limit is $\frac12$ and therefore the series converges. But if you want to compute its sum, you do\begin{align}\sum_{n=1}^\infty\left(\frac1{2^n}+\frac1{3^n}\right)&=\sum_{n=1}^\infty\frac1{2^n}+\sum_{n=1}^\infty\frac1{3^n}\\&=1+\frac12\\&=\frac32.\end{align}Besides, there's no need to use the ratio test. Just use the fact that the sum of two convergent series is convergent too.