text: # Proving $\lim_{n\to\infty}(\frac {a_n}{b_n}) \ne 0 \implies (\sum_{n=1}^\infty a_n =S_1 \iff \sum_{n=1}^\infty b_n =S_2)$
  
  I am having trouble proving the statement below for sequences $a_n\in \mathbb R,b_n\in \mathbb R^+$. Assuming the following limit exists then :
  
  $$x=\lim_{n\to\infty}(\frac {a_n}{b_n}) \ne 0 \implies (\sum_{n=1}^\infty a_n =S_1 \iff \sum_{n=1}^\infty b_n =S_2)$$
  
  Whereas $S_1,S_2$ should simply imply that each of the series converge.
  
  I started out by assuming that $x\ne 0$ and $\sum_{n=1}^\infty a_n$ converges. Now what came to my mind was, since the series converges it implies that $\lim_{n\to\infty}(a_n) = 0$ . Wouldn't that lead to a contradiction at this point though, since I assumed that $x\ne 0$ and if $a_n$ converges to $0$ then it must follow that $x=0$ ,right? Any hints?
  
  • What do you mean with $=S_1$ and $=S_2$?
  – GLay
  Dec 2, 2016 at 14:52
  • @GLay as I said, it should just mean that each of the series converge to some value. Dec 2, 2016 at 14:53
  • And I suppose that the limit can't converge to $\infty$
  – GLay
  Dec 2, 2016 at 14:53
  • @GLay exactly... Dec 2, 2016 at 14:54
  • Have you tried Wikipedia? en.m.wikipedia.org/wiki/Limit_comparison_test
  – GLay
  Dec 2, 2016 at 14:58
  
  There is $N \in \mathbb N$ such that
  
  $\frac{1}{2}|x| \le \frac{|a_n|}{|b_n|} \le \frac{3}{2}|x|$ for $n>N$. Thus
  
  $\frac{1}{2}|x||b_n| \le |a_n| \le \frac{3}{2}|x||b_n|$ for $n>N$.
  
  It follows:
  
  $\sum_{n \ge 1}a_n$ converges iff $\sum_{n \ge 1}b_n$ converges
  
  Hint
  
  Take $\epsilon=\frac{x}{2}$
  
  • use the definition of the limit.
  
  • use comparison test for positive series.
explanation: The document contains a math question and a partial answer or hint to the question. The question is about proving a statement involving limits and series convergence, and the partial answer provides a hint and a step towards solving the problem.
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distilabel_metadata.content: # Proving $\lim_{n\to\infty}(\frac {a_n}{b_n}) \ne 0 \implies (\sum_{n=1}^\infty a_n =S_1 \iff \sum_{n=1}^\infty b_n =S_2)$
  
  I am having trouble proving the statement below for sequences $a_n\in \mathbb R,b_n\in \mathbb R^+$. Assuming the following limit exists then :
  
  $$x=\lim_{n\to\infty}(\frac {a_n}{b_n}) \ne 0 \implies (\sum_{n=1}^\infty a_n =S_1 \iff \sum_{n=1}^\infty b_n =S_2)$$
  
  Whereas $S_1,S_2$ should simply imply that each of the series converge.
  
  I started out by assuming that $x\ne 0$ and $\sum_{n=1}^\infty a_n$ converges. Now what came to my mind was, since the series converges it implies that $\lim_{n\to\infty}(a_n) = 0$ . Wouldn't that lead to a contradiction at this point though, since I assumed that $x\ne 0$ and if $a_n$ converges to $0$ then it must follow that $x=0$ ,right? Any hints?
  
  • What do you mean with $=S_1$ and $=S_2$?
  – GLay
  Dec 2, 2016 at 14:52
  • @GLay as I said, it should just mean that each of the series converge to some value. Dec 2, 2016 at 14:53
  • And I suppose that the limit can't converge to $\infty$
  – GLay
  Dec 2, 2016 at 14:53
  • @GLay exactly... Dec 2, 2016 at 14:54
  • Have you tried Wikipedia? en.m.wikipedia.org/wiki/Limit_comparison_test
  – GLay
  Dec 2, 2016 at 14:58
  
  There is $N \in \mathbb N$ such that
  
  $\frac{1}{2}|x| \le \frac{|a_n|}{|b_n|} \le \frac{3}{2}|x|$ for $n>N$. Thus
  
  $\frac{1}{2}|x||b_n| \le |a_n| \le \frac{3}{2}|x||b_n|$ for $n>N$.
  
  It follows:
  
  $\sum_{n \ge 1}a_n$ converges iff $\sum_{n \ge 1}b_n$ converges
  
  Hint
  
  Take $\epsilon=\frac{x}{2}$
  
  • use the definition of the limit.
  
  • use comparison test for positive series.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a partial answer or hint to the question. The question is about proving a statement involving limits and series convergence, and the partial answer provides a hint and a step towards solving the problem.
  
  Label: question-and-answer