Show convergence but ...

Question

I have to show that $\sum_{n=0}^{\infty}({\frac{1}{2}})^n+({\frac{1}{3}})^n$ is convergent. First I thought I could use the ratio test but the sum start from n=0 and not n=1. How can I then show the convergent?

Answer 1

If two series converges their sum converges $(<\infty$ is sign for convergence ) $$\sum_{n=1}^{\infty} a_n < \infty , \sum_{n=1}^{\infty} b_n < \infty \implies \sum_{n=1}^{\infty} (a_n+b_n) < \infty. $$

Can you prove it?

Answer 2

One more answer:

$a_n:=(1/2)^n+(1/3)^n\le$

$ (1/2)^n +(1/2)^n =2(1/2)^n=:b_n$;

$\sum_{0}^{n}a_n$ converges (Comparison test).