Solution verification: testing the convergence of a sum












1












$begingroup$


I am new to calculus, and I just wanted to check my reasoning re the following:



Given is the following series:



$$sum n^s·e^{-n}, s ge0$$



I was asked to prove that this series converges.



My reasoning was:



$$lim_{nto infty} frac{(n+1)^s}{e^{(n+1)}}·frac{e^n}{n^s}=lim_{nto infty} frac{(n+1)^s}{en^s}=frac {1}{e}$$



As $frac {1}{e}<1$, the series converges.



Is this correct?



Thank you!






calculus sequences-and-series proof-verification







share|cite|improve this question











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  • 2




    $begingroup$
    Yes, it is correct.
    $endgroup$
    – Kavi Rama Murthy
    Jan 8 at 8:25






  • 1




    $begingroup$
    It is correct i would have used root test it is a little quicker
    $endgroup$
    – Milan Stojanovic
    Jan 8 at 8:28
















1












$begingroup$


I am new to calculus, and I just wanted to check my reasoning re the following:



Given is the following series:



$$sum n^s·e^{-n}, s ge0$$



I was asked to prove that this series converges.



My reasoning was:



$$lim_{nto infty} frac{(n+1)^s}{e^{(n+1)}}·frac{e^n}{n^s}=lim_{nto infty} frac{(n+1)^s}{en^s}=frac {1}{e}$$



As $frac {1}{e}<1$, the series converges.



Is this correct?



Thank you!






calculus sequences-and-series proof-verification







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Yes, it is correct.
    $endgroup$
    – Kavi Rama Murthy
    Jan 8 at 8:25






  • 1




    $begingroup$
    It is correct i would have used root test it is a little quicker
    $endgroup$
    – Milan Stojanovic
    Jan 8 at 8:28














1












1








1





$begingroup$


I am new to calculus, and I just wanted to check my reasoning re the following:



Given is the following series:



$$sum n^s·e^{-n}, s ge0$$



I was asked to prove that this series converges.



My reasoning was:



$$lim_{nto infty} frac{(n+1)^s}{e^{(n+1)}}·frac{e^n}{n^s}=lim_{nto infty} frac{(n+1)^s}{en^s}=frac {1}{e}$$



As $frac {1}{e}<1$, the series converges.



Is this correct?



Thank you!






calculus sequences-and-series proof-verification







share|cite|improve this question











$endgroup$




I am new to calculus, and I just wanted to check my reasoning re the following:



Given is the following series:



$$sum n^s·e^{-n}, s ge0$$



I was asked to prove that this series converges.



My reasoning was:



$$lim_{nto infty} frac{(n+1)^s}{e^{(n+1)}}·frac{e^n}{n^s}=lim_{nto infty} frac{(n+1)^s}{en^s}=frac {1}{e}$$



As $frac {1}{e}<1$, the series converges.



Is this correct?



Thank you!






calculus sequences-and-series proof-verification




calculus sequences-and-series proof-verification






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 8 at 8:39









Eevee Trainer

6,1031936




6,1031936










asked Jan 8 at 8:24









daltadalta

1168




1168








  • 2




    $begingroup$
    Yes, it is correct.
    $endgroup$
    – Kavi Rama Murthy
    Jan 8 at 8:25






  • 1




    $begingroup$
    It is correct i would have used root test it is a little quicker
    $endgroup$
    – Milan Stojanovic
    Jan 8 at 8:28














  • 2




    $begingroup$
    Yes, it is correct.
    $endgroup$
    – Kavi Rama Murthy
    Jan 8 at 8:25






  • 1




    $begingroup$
    It is correct i would have used root test it is a little quicker
    $endgroup$
    – Milan Stojanovic
    Jan 8 at 8:28








2




2




$begingroup$
Yes, it is correct.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 8:25




$begingroup$
Yes, it is correct.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 8:25




1




1




$begingroup$
It is correct i would have used root test it is a little quicker
$endgroup$
– Milan Stojanovic
Jan 8 at 8:28




$begingroup$
It is correct i would have used root test it is a little quicker
$endgroup$
– Milan Stojanovic
Jan 8 at 8:28










1 Answer
1






active

oldest

votes


















1












$begingroup$

Your proof is definitely valid as an application of the ratio test.



One minor nitpick: it would've been more appropriate to indicate the index of summation, i.e.



$$sum_{n=0}^infty ;;; text{instead of} ;;; sum$$



While it's okay to use the latter when the indices or the space over which you're summing is well understood, it's not necessarily true here, in my opinion. But this may well be just a personal issue since I struggled for a sec in figuring out which variable you were summing over.



(For completeness' sake I assume it was something like the former, an infinite summation starting at some finite index, i.e. $0, 1,$ etc., and was a summation over $n$.)






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    thank you! strangely enough, that's the way it appears in the exercise, so I didn't change it...
    $endgroup$
    – dalta
    Jan 8 at 8:42










  • $begingroup$
    Huh, that's interesting. I mostly point it out because the main thing that could make the textbook's answer unusual is if it were understood be a finite sum (say from $n=1$ to $n=100$ or something). Of course that's probably not the case. It's probably just assumed it's infinite and starts at a finite index then, since a lot of these problems work like that. So there shouldn't be an issue I suppose.
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:43










  • $begingroup$
    (Post-script: Of course, it'll never hurt to check with your teacher/professor if you feel unsure. :p)
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:44











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









1












$begingroup$

Your proof is definitely valid as an application of the ratio test.



One minor nitpick: it would've been more appropriate to indicate the index of summation, i.e.



$$sum_{n=0}^infty ;;; text{instead of} ;;; sum$$



While it's okay to use the latter when the indices or the space over which you're summing is well understood, it's not necessarily true here, in my opinion. But this may well be just a personal issue since I struggled for a sec in figuring out which variable you were summing over.



(For completeness' sake I assume it was something like the former, an infinite summation starting at some finite index, i.e. $0, 1,$ etc., and was a summation over $n$.)






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    thank you! strangely enough, that's the way it appears in the exercise, so I didn't change it...
    $endgroup$
    – dalta
    Jan 8 at 8:42










  • $begingroup$
    Huh, that's interesting. I mostly point it out because the main thing that could make the textbook's answer unusual is if it were understood be a finite sum (say from $n=1$ to $n=100$ or something). Of course that's probably not the case. It's probably just assumed it's infinite and starts at a finite index then, since a lot of these problems work like that. So there shouldn't be an issue I suppose.
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:43










  • $begingroup$
    (Post-script: Of course, it'll never hurt to check with your teacher/professor if you feel unsure. :p)
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:44
















1












$begingroup$

Your proof is definitely valid as an application of the ratio test.



One minor nitpick: it would've been more appropriate to indicate the index of summation, i.e.



$$sum_{n=0}^infty ;;; text{instead of} ;;; sum$$



While it's okay to use the latter when the indices or the space over which you're summing is well understood, it's not necessarily true here, in my opinion. But this may well be just a personal issue since I struggled for a sec in figuring out which variable you were summing over.



(For completeness' sake I assume it was something like the former, an infinite summation starting at some finite index, i.e. $0, 1,$ etc., and was a summation over $n$.)






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    thank you! strangely enough, that's the way it appears in the exercise, so I didn't change it...
    $endgroup$
    – dalta
    Jan 8 at 8:42










  • $begingroup$
    Huh, that's interesting. I mostly point it out because the main thing that could make the textbook's answer unusual is if it were understood be a finite sum (say from $n=1$ to $n=100$ or something). Of course that's probably not the case. It's probably just assumed it's infinite and starts at a finite index then, since a lot of these problems work like that. So there shouldn't be an issue I suppose.
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:43










  • $begingroup$
    (Post-script: Of course, it'll never hurt to check with your teacher/professor if you feel unsure. :p)
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:44














1












1








1





$begingroup$

Your proof is definitely valid as an application of the ratio test.



One minor nitpick: it would've been more appropriate to indicate the index of summation, i.e.



$$sum_{n=0}^infty ;;; text{instead of} ;;; sum$$



While it's okay to use the latter when the indices or the space over which you're summing is well understood, it's not necessarily true here, in my opinion. But this may well be just a personal issue since I struggled for a sec in figuring out which variable you were summing over.



(For completeness' sake I assume it was something like the former, an infinite summation starting at some finite index, i.e. $0, 1,$ etc., and was a summation over $n$.)






share|cite|improve this answer









$endgroup$



Your proof is definitely valid as an application of the ratio test.



One minor nitpick: it would've been more appropriate to indicate the index of summation, i.e.



$$sum_{n=0}^infty ;;; text{instead of} ;;; sum$$



While it's okay to use the latter when the indices or the space over which you're summing is well understood, it's not necessarily true here, in my opinion. But this may well be just a personal issue since I struggled for a sec in figuring out which variable you were summing over.



(For completeness' sake I assume it was something like the former, an infinite summation starting at some finite index, i.e. $0, 1,$ etc., and was a summation over $n$.)







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 8 at 8:37









Eevee TrainerEevee Trainer

6,1031936




6,1031936








  • 1




    $begingroup$
    thank you! strangely enough, that's the way it appears in the exercise, so I didn't change it...
    $endgroup$
    – dalta
    Jan 8 at 8:42










  • $begingroup$
    Huh, that's interesting. I mostly point it out because the main thing that could make the textbook's answer unusual is if it were understood be a finite sum (say from $n=1$ to $n=100$ or something). Of course that's probably not the case. It's probably just assumed it's infinite and starts at a finite index then, since a lot of these problems work like that. So there shouldn't be an issue I suppose.
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:43










  • $begingroup$
    (Post-script: Of course, it'll never hurt to check with your teacher/professor if you feel unsure. :p)
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:44














  • 1




    $begingroup$
    thank you! strangely enough, that's the way it appears in the exercise, so I didn't change it...
    $endgroup$
    – dalta
    Jan 8 at 8:42










  • $begingroup$
    Huh, that's interesting. I mostly point it out because the main thing that could make the textbook's answer unusual is if it were understood be a finite sum (say from $n=1$ to $n=100$ or something). Of course that's probably not the case. It's probably just assumed it's infinite and starts at a finite index then, since a lot of these problems work like that. So there shouldn't be an issue I suppose.
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:43










  • $begingroup$
    (Post-script: Of course, it'll never hurt to check with your teacher/professor if you feel unsure. :p)
    $endgroup$
    – Eevee Trainer
    Jan 8 at 8:44








1




1




$begingroup$
thank you! strangely enough, that's the way it appears in the exercise, so I didn't change it...
$endgroup$
– dalta
Jan 8 at 8:42




$begingroup$
thank you! strangely enough, that's the way it appears in the exercise, so I didn't change it...
$endgroup$
– dalta
Jan 8 at 8:42












$begingroup$
Huh, that's interesting. I mostly point it out because the main thing that could make the textbook's answer unusual is if it were understood be a finite sum (say from $n=1$ to $n=100$ or something). Of course that's probably not the case. It's probably just assumed it's infinite and starts at a finite index then, since a lot of these problems work like that. So there shouldn't be an issue I suppose.
$endgroup$
– Eevee Trainer
Jan 8 at 8:43




$begingroup$
Huh, that's interesting. I mostly point it out because the main thing that could make the textbook's answer unusual is if it were understood be a finite sum (say from $n=1$ to $n=100$ or something). Of course that's probably not the case. It's probably just assumed it's infinite and starts at a finite index then, since a lot of these problems work like that. So there shouldn't be an issue I suppose.
$endgroup$
– Eevee Trainer
Jan 8 at 8:43












$begingroup$
(Post-script: Of course, it'll never hurt to check with your teacher/professor if you feel unsure. :p)
$endgroup$
– Eevee Trainer
Jan 8 at 8:44




$begingroup$
(Post-script: Of course, it'll never hurt to check with your teacher/professor if you feel unsure. :p)
$endgroup$
– Eevee Trainer
Jan 8 at 8:44


















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