Show that $frac{3^n}{ncdot5^n}$ converges absolutely [on hold]
I have to show that $$frac{3^n}{ncdot5^n} $$ converges absolutely. This time I think that I actually got it right, but I would like very sure that I got it right.
My result: with the ratio test it is $frac{3}{10}$ which is $<1$, and therefore the series converges absolutely.
Full result on my paper: 
analysis proof-verification convergence alternative-proof absolute-convergence
edited Dec 2 at 16:05
Jam
4,94811431
asked Dec 2 at 16:00
SacredScout
487
put on hold as off-topic by RRL, NCh, Eevee Trainer, Leucippus, Holo yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, NCh, Eevee Trainer, Leucippus, Holo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
I have to show that $$frac{3^n}{ncdot5^n} $$ converges absolutely. This time I think that I actually got it right, but I would like very sure that I got it right.
My result: with the ratio test it is $frac{3}{10}$ which is $<1$, and therefore the series converges absolutely.
Full result on my paper:
analysis proof-verification convergence alternative-proof absolute-convergence
edited Dec 2 at 16:05
Jam
4,94811431
asked Dec 2 at 16:00
SacredScout
487
put on hold as off-topic by RRL, NCh, Eevee Trainer, Leucippus, Holo yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, NCh, Eevee Trainer, Leucippus, Holo
If this question can be reworded to fit the rules in the help center, please edit the question.
2
Can you please explain the equality $frac{3n}{5n+5}=frac3{5+5}$?
– José Carlos Santos
Dec 2 at 16:01
$frac{3n}{5n+5}$=$frac{3}{5}*frac{n}{n+5}$ or not?
– SacredScout
Dec 2 at 16:05
1
$frac{3}{5n+5}=frac35 cdot frac{n}{n+1}$.
– Siong Thye Goh
Dec 2 at 16:07
1
@SacredScout No. It is false for every single natural $n$.
– José Carlos Santos
Dec 2 at 16:14
add a comment |
I have to show that $$frac{3^n}{ncdot5^n} $$ converges absolutely. This time I think that I actually got it right, but I would like very sure that I got it right.
My result: with the ratio test it is $frac{3}{10}$ which is $<1$, and therefore the series converges absolutely.
Full result on my paper:
analysis proof-verification convergence alternative-proof absolute-convergence
edited Dec 2 at 16:05
Jam
4,94811431
asked Dec 2 at 16:00
SacredScout
487
I have to show that $$frac{3^n}{ncdot5^n} $$ converges absolutely. This time I think that I actually got it right, but I would like very sure that I got it right.
My result: with the ratio test it is $frac{3}{10}$ which is $<1$, and therefore the series converges absolutely.
Full result on my paper:
analysis proof-verification convergence alternative-proof absolute-convergence
analysis proof-verification convergence alternative-proof absolute-convergence
edited Dec 2 at 16:05
Jam
4,94811431
asked Dec 2 at 16:00
SacredScout
487
edited Dec 2 at 16:05
Jam
4,94811431
asked Dec 2 at 16:00
SacredScout
487
edited Dec 2 at 16:05
Jam
4,94811431
edited Dec 2 at 16:05
Jam
4,94811431
edited Dec 2 at 16:05
Jam
4,94811431
4,94811431
asked Dec 2 at 16:00
SacredScout
487
asked Dec 2 at 16:00
SacredScout
487
asked Dec 2 at 16:00
SacredScout
487
487
put on hold as off-topic by RRL, NCh, Eevee Trainer, Leucippus, Holo yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, NCh, Eevee Trainer, Leucippus, Holo
If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as off-topic by RRL, NCh, Eevee Trainer, Leucippus, Holo yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, NCh, Eevee Trainer, Leucippus, Holo
If this question can be reworded to fit the rules in the help center, please edit the question.
2
Can you please explain the equality $frac{3n}{5n+5}=frac3{5+5}$?
– José Carlos Santos
Dec 2 at 16:01
$frac{3n}{5n+5}$=$frac{3}{5}*frac{n}{n+5}$ or not?
– SacredScout
Dec 2 at 16:05
1
$frac{3}{5n+5}=frac35 cdot frac{n}{n+1}$.
– Siong Thye Goh
Dec 2 at 16:07
1
@SacredScout No. It is false for every single natural $n$.
– José Carlos Santos
Dec 2 at 16:14
add a comment |
2
Can you please explain the equality $frac{3n}{5n+5}=frac3{5+5}$?
– José Carlos Santos
Dec 2 at 16:01
$frac{3n}{5n+5}$=$frac{3}{5}*frac{n}{n+5}$ or not?
– SacredScout
Dec 2 at 16:05
1
$frac{3}{5n+5}=frac35 cdot frac{n}{n+1}$.
– Siong Thye Goh
Dec 2 at 16:07
1
@SacredScout No. It is false for every single natural $n$.
– José Carlos Santos
Dec 2 at 16:14
2
2
Can you please explain the equality $frac{3n}{5n+5}=frac3{5+5}$?
– José Carlos Santos
Dec 2 at 16:01
Can you please explain the equality $frac{3n}{5n+5}=frac3{5+5}$?
– José Carlos Santos
Dec 2 at 16:01
$frac{3n}{5n+5}$=$frac{3}{5}*frac{n}{n+5}$ or not?
– SacredScout
Dec 2 at 16:05
$frac{3n}{5n+5}$=$frac{3}{5}*frac{n}{n+5}$ or not?
– SacredScout
Dec 2 at 16:05
1
1
$frac{3}{5n+5}=frac35 cdot frac{n}{n+1}$.
– Siong Thye Goh
Dec 2 at 16:07
$frac{3}{5n+5}=frac35 cdot frac{n}{n+1}$.
– Siong Thye Goh
Dec 2 at 16:07
1
1
@SacredScout No. It is false for every single natural $n$.
– José Carlos Santos
Dec 2 at 16:14
@SacredScout No. It is false for every single natural $n$.
– José Carlos Santos
Dec 2 at 16:14
add a comment |
2 Answers
2
active
oldest
votes
Yes it converges indeed more simply
$$frac{3^n}{ncdot 5^n} le frac{3^n}{5^n}=left(frac53right)^n $$
then refer to geometric series with $r=3/5$.
As an alternative also by ratio test
$$sqrt[n]{frac{3^n}{ncdot 5^n}}tofrac35<1$$
we can see that the series converges.
Note that since the series is with positive terms, we don't need to say that it converges absolutely, it suffices to observe simply that it converges.
answered Dec 2 at 16:03
gimusi
1
What would I exactly have to say to proof it by referring to the geometric series?
– SacredScout
Dec 2 at 16:08
1
Because the geometric series converges since |r|<1.
– gimusi
Dec 2 at 17:54
add a comment |
You almost got it,
$$lim_{nto infty}frac{a_{n+1}}{a_n}= lim_{n to infty}frac{3n}{5n+5}=lim_{n to infty}frac{3}{5+frac5{n}}=frac35 < 1$$
answered Dec 2 at 16:05
Siong Thye Goh
99.1k1464117
Of course, that is how Wolfram Alpha did it I suppose. I was wondering why I got $frac{3}{10$. Thanks! $5/n$ can simply be ignored because of lim n to infinity, correct? Also: Thank you for answering nearly every post I made so far! Amazing, really!
– SacredScout
Dec 2 at 16:09
1
We have $frac5n to 0$.
– Siong Thye Goh
Dec 2 at 16:10
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Yes it converges indeed more simply
$$frac{3^n}{ncdot 5^n} le frac{3^n}{5^n}=left(frac53right)^n $$
then refer to geometric series with $r=3/5$.
As an alternative also by ratio test
$$sqrt[n]{frac{3^n}{ncdot 5^n}}tofrac35<1$$
we can see that the series converges.
Note that since the series is with positive terms, we don't need to say that it converges absolutely, it suffices to observe simply that it converges.
answered Dec 2 at 16:03
gimusi
1
What would I exactly have to say to proof it by referring to the geometric series?
– SacredScout
Dec 2 at 16:08
1
Because the geometric series converges since |r|<1.
– gimusi
Dec 2 at 17:54
add a comment |
Yes it converges indeed more simply
$$frac{3^n}{ncdot 5^n} le frac{3^n}{5^n}=left(frac53right)^n $$
then refer to geometric series with $r=3/5$.
As an alternative also by ratio test
$$sqrt[n]{frac{3^n}{ncdot 5^n}}tofrac35<1$$
we can see that the series converges.
Note that since the series is with positive terms, we don't need to say that it converges absolutely, it suffices to observe simply that it converges.
answered Dec 2 at 16:03
gimusi
1
What would I exactly have to say to proof it by referring to the geometric series?
– SacredScout
Dec 2 at 16:08
1
Because the geometric series converges since |r|<1.
– gimusi
Dec 2 at 17:54
add a comment |
Yes it converges indeed more simply
$$frac{3^n}{ncdot 5^n} le frac{3^n}{5^n}=left(frac53right)^n $$
then refer to geometric series with $r=3/5$.
As an alternative also by ratio test
$$sqrt[n]{frac{3^n}{ncdot 5^n}}tofrac35<1$$
we can see that the series converges.
Note that since the series is with positive terms, we don't need to say that it converges absolutely, it suffices to observe simply that it converges.
answered Dec 2 at 16:03
gimusi
1
Yes it converges indeed more simply
$$frac{3^n}{ncdot 5^n} le frac{3^n}{5^n}=left(frac53right)^n $$
then refer to geometric series with $r=3/5$.
As an alternative also by ratio test
$$sqrt[n]{frac{3^n}{ncdot 5^n}}tofrac35<1$$
we can see that the series converges.
Note that since the series is with positive terms, we don't need to say that it converges absolutely, it suffices to observe simply that it converges.
answered Dec 2 at 16:03
gimusi
1
edited Dec 2 at 18:11
answered Dec 2 at 16:03
gimusi
1
answered Dec 2 at 16:03
gimusi
1
answered Dec 2 at 16:03
gimusi
1
1
What would I exactly have to say to proof it by referring to the geometric series?
– SacredScout
Dec 2 at 16:08
1
Because the geometric series converges since |r|<1.
– gimusi
Dec 2 at 17:54
add a comment |
What would I exactly have to say to proof it by referring to the geometric series?
– SacredScout
Dec 2 at 16:08
1
Because the geometric series converges since |r|<1.
– gimusi
Dec 2 at 17:54
What would I exactly have to say to proof it by referring to the geometric series?
– SacredScout
Dec 2 at 16:08
What would I exactly have to say to proof it by referring to the geometric series?
– SacredScout
Dec 2 at 16:08
1
1
Because the geometric series converges since |r|<1.
– gimusi
Dec 2 at 17:54
Because the geometric series converges since |r|<1.
– gimusi
Dec 2 at 17:54
add a comment |
You almost got it,
$$lim_{nto infty}frac{a_{n+1}}{a_n}= lim_{n to infty}frac{3n}{5n+5}=lim_{n to infty}frac{3}{5+frac5{n}}=frac35 < 1$$
answered Dec 2 at 16:05
Siong Thye Goh
99.1k1464117
Of course, that is how Wolfram Alpha did it I suppose. I was wondering why I got $frac{3}{10$. Thanks! $5/n$ can simply be ignored because of lim n to infinity, correct? Also: Thank you for answering nearly every post I made so far! Amazing, really!
– SacredScout
Dec 2 at 16:09
1
We have $frac5n to 0$.
– Siong Thye Goh
Dec 2 at 16:10
add a comment |
You almost got it,
$$lim_{nto infty}frac{a_{n+1}}{a_n}= lim_{n to infty}frac{3n}{5n+5}=lim_{n to infty}frac{3}{5+frac5{n}}=frac35 < 1$$
answered Dec 2 at 16:05
Siong Thye Goh
99.1k1464117
Of course, that is how Wolfram Alpha did it I suppose. I was wondering why I got $frac{3}{10$. Thanks! $5/n$ can simply be ignored because of lim n to infinity, correct? Also: Thank you for answering nearly every post I made so far! Amazing, really!
– SacredScout
Dec 2 at 16:09
1
We have $frac5n to 0$.
– Siong Thye Goh
Dec 2 at 16:10
add a comment |
You almost got it,
$$lim_{nto infty}frac{a_{n+1}}{a_n}= lim_{n to infty}frac{3n}{5n+5}=lim_{n to infty}frac{3}{5+frac5{n}}=frac35 < 1$$
answered Dec 2 at 16:05
Siong Thye Goh
99.1k1464117
You almost got it,
$$lim_{nto infty}frac{a_{n+1}}{a_n}= lim_{n to infty}frac{3n}{5n+5}=lim_{n to infty}frac{3}{5+frac5{n}}=frac35 < 1$$
answered Dec 2 at 16:05
Siong Thye Goh
99.1k1464117
answered Dec 2 at 16:05
Siong Thye Goh
99.1k1464117
answered Dec 2 at 16:05
Siong Thye Goh
99.1k1464117
answered Dec 2 at 16:05
Siong Thye Goh
99.1k1464117
99.1k1464117
Of course, that is how Wolfram Alpha did it I suppose. I was wondering why I got $frac{3}{10$. Thanks! $5/n$ can simply be ignored because of lim n to infinity, correct? Also: Thank you for answering nearly every post I made so far! Amazing, really!
– SacredScout
Dec 2 at 16:09
1
We have $frac5n to 0$.
– Siong Thye Goh
Dec 2 at 16:10
add a comment |
Of course, that is how Wolfram Alpha did it I suppose. I was wondering why I got $frac{3}{10$. Thanks! $5/n$ can simply be ignored because of lim n to infinity, correct? Also: Thank you for answering nearly every post I made so far! Amazing, really!
– SacredScout
Dec 2 at 16:09
1
We have $frac5n to 0$.
– Siong Thye Goh
Dec 2 at 16:10
Of course, that is how Wolfram Alpha did it I suppose. I was wondering why I got $frac{3}{10$. Thanks! $5/n$ can simply be ignored because of lim n to infinity, correct? Also: Thank you for answering nearly every post I made so far! Amazing, really!
– SacredScout
Dec 2 at 16:09
Of course, that is how Wolfram Alpha did it I suppose. I was wondering why I got $frac{3}{10$. Thanks! $5/n$ can simply be ignored because of lim n to infinity, correct? Also: Thank you for answering nearly every post I made so far! Amazing, really!
– SacredScout
Dec 2 at 16:09
1
1
We have $frac5n to 0$.
– Siong Thye Goh
Dec 2 at 16:10
We have $frac5n to 0$.
– Siong Thye Goh
Dec 2 at 16:10
add a comment |
2
Can you please explain the equality $frac{3n}{5n+5}=frac3{5+5}$?
– José Carlos Santos
Dec 2 at 16:01
$frac{3n}{5n+5}$=$frac{3}{5}*frac{n}{n+5}$ or not?
– SacredScout
Dec 2 at 16:05
1
$frac{3}{5n+5}=frac35 cdot frac{n}{n+1}$.
– Siong Thye Goh
Dec 2 at 16:07
1
@SacredScout No. It is false for every single natural $n$.
– José Carlos Santos
Dec 2 at 16:14