Counterexamples related to a convergent positive series












0















Let ${a_n}$ be a sequence such that $a_n > 0$ for all $n ≥ 1$ and $sum_1^infty$$a_n$
converges.



Give
counterexamples to the following claims where $b_n$ =
$a_{n+1}$/$a_n$



(a) $a_n ≤ 1$ for all $n ≥ 1$.



(b) The sequence ${a_n}$ is non-increasing.



(c) $lim_{nto infty}b_n$ exists.



(d) If $lim_{nto infty}b_n$ exists, then $lim_{nto infty}b_n < 1$.



(e) The sequence ${b_n}$ is bounded.



(f) If $limsup_{nto infty}b_n$ exists, then $limsup_{nto infty}b_nleq 1$.




My attempt: $2/n^3$ works for (a) and (d). I would really appreciate help for the other counter-examples.






sequences-and-series







share|cite|improve this question
























  • I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
    – childishsadbino
    Dec 11 '18 at 6:38










  • You should add these to your post.
    – xbh
    Dec 11 '18 at 6:44










  • $a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
    – mathworker21
    Dec 11 '18 at 6:47










  • @mathworker21 no, we have to come up with counter-examples for these statements.
    – childishsadbino
    Dec 11 '18 at 6:50










  • @childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
    – mathworker21
    Dec 11 '18 at 6:51


















0















Let ${a_n}$ be a sequence such that $a_n > 0$ for all $n ≥ 1$ and $sum_1^infty$$a_n$
converges.



Give
counterexamples to the following claims where $b_n$ =
$a_{n+1}$/$a_n$



(a) $a_n ≤ 1$ for all $n ≥ 1$.



(b) The sequence ${a_n}$ is non-increasing.



(c) $lim_{nto infty}b_n$ exists.



(d) If $lim_{nto infty}b_n$ exists, then $lim_{nto infty}b_n < 1$.



(e) The sequence ${b_n}$ is bounded.



(f) If $limsup_{nto infty}b_n$ exists, then $limsup_{nto infty}b_nleq 1$.




My attempt: $2/n^3$ works for (a) and (d). I would really appreciate help for the other counter-examples.






sequences-and-series







share|cite|improve this question
























  • I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
    – childishsadbino
    Dec 11 '18 at 6:38










  • You should add these to your post.
    – xbh
    Dec 11 '18 at 6:44










  • $a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
    – mathworker21
    Dec 11 '18 at 6:47










  • @mathworker21 no, we have to come up with counter-examples for these statements.
    – childishsadbino
    Dec 11 '18 at 6:50










  • @childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
    – mathworker21
    Dec 11 '18 at 6:51
















0












0








0








Let ${a_n}$ be a sequence such that $a_n > 0$ for all $n ≥ 1$ and $sum_1^infty$$a_n$
converges.



Give
counterexamples to the following claims where $b_n$ =
$a_{n+1}$/$a_n$



(a) $a_n ≤ 1$ for all $n ≥ 1$.



(b) The sequence ${a_n}$ is non-increasing.



(c) $lim_{nto infty}b_n$ exists.



(d) If $lim_{nto infty}b_n$ exists, then $lim_{nto infty}b_n < 1$.



(e) The sequence ${b_n}$ is bounded.



(f) If $limsup_{nto infty}b_n$ exists, then $limsup_{nto infty}b_nleq 1$.




My attempt: $2/n^3$ works for (a) and (d). I would really appreciate help for the other counter-examples.






sequences-and-series







share|cite|improve this question
















Let ${a_n}$ be a sequence such that $a_n > 0$ for all $n ≥ 1$ and $sum_1^infty$$a_n$
converges.



Give
counterexamples to the following claims where $b_n$ =
$a_{n+1}$/$a_n$



(a) $a_n ≤ 1$ for all $n ≥ 1$.



(b) The sequence ${a_n}$ is non-increasing.



(c) $lim_{nto infty}b_n$ exists.



(d) If $lim_{nto infty}b_n$ exists, then $lim_{nto infty}b_n < 1$.



(e) The sequence ${b_n}$ is bounded.



(f) If $limsup_{nto infty}b_n$ exists, then $limsup_{nto infty}b_nleq 1$.




My attempt: $2/n^3$ works for (a) and (d). I would really appreciate help for the other counter-examples.






sequences-and-series




sequences-and-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 11 '18 at 7:03









Robert Z

93.4k1061132




93.4k1061132










asked Dec 11 '18 at 6:25









childishsadbino

1148




1148












  • I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
    – childishsadbino
    Dec 11 '18 at 6:38










  • You should add these to your post.
    – xbh
    Dec 11 '18 at 6:44










  • $a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
    – mathworker21
    Dec 11 '18 at 6:47










  • @mathworker21 no, we have to come up with counter-examples for these statements.
    – childishsadbino
    Dec 11 '18 at 6:50










  • @childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
    – mathworker21
    Dec 11 '18 at 6:51




















  • I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
    – childishsadbino
    Dec 11 '18 at 6:38










  • You should add these to your post.
    – xbh
    Dec 11 '18 at 6:44










  • $a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
    – mathworker21
    Dec 11 '18 at 6:47










  • @mathworker21 no, we have to come up with counter-examples for these statements.
    – childishsadbino
    Dec 11 '18 at 6:50










  • @childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
    – mathworker21
    Dec 11 '18 at 6:51


















I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
– childishsadbino
Dec 11 '18 at 6:38




I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
– childishsadbino
Dec 11 '18 at 6:38












You should add these to your post.
– xbh
Dec 11 '18 at 6:44




You should add these to your post.
– xbh
Dec 11 '18 at 6:44












$a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
– mathworker21
Dec 11 '18 at 6:47




$a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
– mathworker21
Dec 11 '18 at 6:47












@mathworker21 no, we have to come up with counter-examples for these statements.
– childishsadbino
Dec 11 '18 at 6:50




@mathworker21 no, we have to come up with counter-examples for these statements.
– childishsadbino
Dec 11 '18 at 6:50












@childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
– mathworker21
Dec 11 '18 at 6:51






@childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
– mathworker21
Dec 11 '18 at 6:51












2 Answers
2






active

oldest

votes


















1














Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?






share|cite|improve this answer























  • I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    – Robert Z
    Dec 11 '18 at 7:17












  • Makes sense now! Thank you so much!
    – childishsadbino
    Dec 11 '18 at 7:21



















1














For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$






share|cite|improve this answer





















  • This should serve as a counter-example for (e) as well, right?
    – childishsadbino
    Dec 11 '18 at 7:02










  • Yeah.${{{{{{}}}}}}$
    – xbh
    Dec 11 '18 at 7:02











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









1














Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?






share|cite|improve this answer























  • I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    – Robert Z
    Dec 11 '18 at 7:17












  • Makes sense now! Thank you so much!
    – childishsadbino
    Dec 11 '18 at 7:21
















1














Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?






share|cite|improve this answer























  • I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    – Robert Z
    Dec 11 '18 at 7:17












  • Makes sense now! Thank you so much!
    – childishsadbino
    Dec 11 '18 at 7:21














1












1








1






Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?






share|cite|improve this answer














Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 11 '18 at 6:58

























answered Dec 11 '18 at 6:52









Robert Z

93.4k1061132




93.4k1061132












  • I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    – Robert Z
    Dec 11 '18 at 7:17












  • Makes sense now! Thank you so much!
    – childishsadbino
    Dec 11 '18 at 7:21


















  • I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    – Robert Z
    Dec 11 '18 at 7:17












  • Makes sense now! Thank you so much!
    – childishsadbino
    Dec 11 '18 at 7:21
















I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
– childishsadbino
Dec 11 '18 at 7:14




I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
– childishsadbino
Dec 11 '18 at 7:14




1




1




@childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
– Robert Z
Dec 11 '18 at 7:17






@childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
– Robert Z
Dec 11 '18 at 7:17














Makes sense now! Thank you so much!
– childishsadbino
Dec 11 '18 at 7:21




Makes sense now! Thank you so much!
– childishsadbino
Dec 11 '18 at 7:21











1














For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$






share|cite|improve this answer





















  • This should serve as a counter-example for (e) as well, right?
    – childishsadbino
    Dec 11 '18 at 7:02










  • Yeah.${{{{{{}}}}}}$
    – xbh
    Dec 11 '18 at 7:02
















1














For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$






share|cite|improve this answer





















  • This should serve as a counter-example for (e) as well, right?
    – childishsadbino
    Dec 11 '18 at 7:02










  • Yeah.${{{{{{}}}}}}$
    – xbh
    Dec 11 '18 at 7:02














1












1








1






For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$






share|cite|improve this answer












For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 11 '18 at 6:59









xbh

5,6751522




5,6751522












  • This should serve as a counter-example for (e) as well, right?
    – childishsadbino
    Dec 11 '18 at 7:02










  • Yeah.${{{{{{}}}}}}$
    – xbh
    Dec 11 '18 at 7:02


















  • This should serve as a counter-example for (e) as well, right?
    – childishsadbino
    Dec 11 '18 at 7:02










  • Yeah.${{{{{{}}}}}}$
    – xbh
    Dec 11 '18 at 7:02
















This should serve as a counter-example for (e) as well, right?
– childishsadbino
Dec 11 '18 at 7:02




This should serve as a counter-example for (e) as well, right?
– childishsadbino
Dec 11 '18 at 7:02












Yeah.${{{{{{}}}}}}$
– xbh
Dec 11 '18 at 7:02




Yeah.${{{{{{}}}}}}$
– xbh
Dec 11 '18 at 7:02


















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