Is $a_{2n} < a_n$?

Question

Suppose we have 2 convergent series $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} a_{2n}$. Does that mean that $a_{2n} \leq a_n$?

Answer 1

No. Consider $a_n = 1$ if $n=2$, and $a_n =0$ for all $n \neq 2$. Then your two series converge, but $a_2 > a_1$.

Answer 2

Certainly not. Try $$ a_n = \cases{1/2^n & if $n$ is even\cr 0 & if $n$ is odd\cr} $$