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Proof

(related to Proposition: Raabe's Test)

• By hypothesis, $\sum_{n=0}^\infty a_n$ is an infinite series.
• Assume, for a fixed positive number $\beta > 1$ there exists an index $N$ such that $a_n\neq 0$ and $\left|\frac{a_{n+1}}{a_n}\right|\le 1-\frac{\beta}{n}$ for all $n\ge N.$
  • This is equivalent to $n|a_{n+1}|\le n|a_{n}|-\beta|a_n|$ for all $n\ge N.$
  • Thus, $n|a_{n+1}|\le n|a_{n}|+|a_{n}|-\beta|a_n|-|a_n|$ and therefore $(\beta-1)|a_n|\le (n-1)|a_n|-n|a_{n+1}|$ for all $n\ge N.$
  • Since $\beta > 1$ we get $|a_n|<(n-1)|a_n|-n|a_{n+1}|$ and by rules of calculations with inequalities $0<|a_n|-|a_{n+1}|$ which is equivalent to $|a_{n+1}| < |a_n|$.
  • It follows that the sequence $(|a_n|)_{n\in\mathbb N}$ is monotonically decreasing.
  • On the other hand, the sequence is bounded, since $|a_n| > 0.$
  • Because every bounded monontonic sequence is convergent, $(|a_n|)_{n\in\mathbb N}$ is a convergent sequence.
  • Set $b_n:=(n-1)|a_n|-n|a_{n+1}|$ for all $n\ge N.$
  • Then the partial sums $\sum_{k=0}^Kb_k$ are, in fact, telescoping series and thus, the infinite series $\sum_{k=0}^\infty b_k$ is convergent.
  • Since $|a_n|\le\frac{b_n}{\beta-1}$, the infinite series $\sum_{n=0}^\infty a_n$ is absolutely convergent by the direct comparison test for absolutely convergent series.
• Now, assume there is an index $N$ such that $a_n\neq 0$ and $\left|\frac{a_{n+1}}{a_n}\right|\ge 1-\frac{1}{n}$ for all $n\ge N.$
  • This is equivalent to $n|a_{n+1}|\ge (n-1)|a_n|$ for all $n\ge N.$
  • This means that the sequence $(n|a_{n+1}|)$ is a positive-valued, monotonically increasing sequence, and therefore, its squence members exceed finally a positive bound $\alpha > 0.$
  • It follows that $\frac{|a_{n+1}|}\alpha\ge \frac 1n.$
  • Since the harmonic series diverges, it follows that the infinite series $\frac{1}{\alpha}\sum_{n=0}^\infty a_n$ diverges, again by the direct comparison test.
  • Therefore, except of the constant $\frac{1}{\alpha}$, the infinite series $\sum_{n=0}^\infty a_n$ diverges.
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References

Bibliography

1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition