# Existence of a specific reordering bijection

Please consider a bijection $g:\mathbb{N}\rightarrow\mathbb{N}$ with following properties:

1. For all real series $(a_n)_{n\geq1}$, convergence of $\sum_{n=1}^{\infty}a_n$ implies convergence of $\sum_{n=1}^{\infty}a_{g(n)}$

2. Exist at least one real series $(c_n)_{c\geq1}$, that $\sum_{n=1}^{\infty} c_n$ diverge, but $\sum_{n=1}^{\infty}c_{g(n)}$ converge.

If such bijection exist?

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A detailed answer to your question can be found on the paper "Creating More Convergent Series" by Steven Krantz and Jeffery McNeal, which can be found here. – Matemáticos Chibchas Jan 10 '13 at 20:32
@MatemáticosChibchas I guess this qualifies to be turned into an answer ... – Hagen von Eitzen Jan 10 '13 at 21:37