Mixing
Find the limit $limlimits_{n to infty}sum_{k=1}^n frac{sqrt{4n+5k}}{n sqrt{n}} $
$begingroup$
I got stuck on the following limit:
$limlimits_{n to infty}5sum_{k=1}^n frac{sqrt{4n+5k}}{n sqrt{n}} $
The expected answer is $frac{38}{15}.$ Any hint is appreciated. Thanks
EDIT: The task is from this polish textbook (http://lucc.pl/inf/analiza_1/gewert_skoczylas__analiza_matematyczna_1__definicje_twierdzenia_wzory.pdf). It's on page 166 of the book (ex. 8.4.3 h). I want to compute this using the above fact. It's a book for first year students.
limits summation
asked Jan 25 at 20:09
The CatThe Cat
33815
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add a comment |
$begingroup$
I got stuck on the following limit:
$limlimits_{n to infty}5sum_{k=1}^n frac{sqrt{4n+5k}}{n sqrt{n}} $
The expected answer is $frac{38}{15}.$ Any hint is appreciated. Thanks
EDIT: The task is from this polish textbook (http://lucc.pl/inf/analiza_1/gewert_skoczylas__analiza_matematyczna_1__definicje_twierdzenia_wzory.pdf). It's on page 166 of the book (ex. 8.4.3 h). I want to compute this using the above fact. It's a book for first year students.
limits summation
asked Jan 25 at 20:09
The CatThe Cat
33815
$endgroup$
$begingroup$
If you rearrange the terms a bit, this is a Riemann sum. So it converges to some integral. Can you take it from there?
$endgroup$
– Stefan Lafon
Jan 25 at 20:12
$begingroup$
Well this task here is from a prep-book to college. I've just started calculating definite integrals by definition, and in order to do the exercise i need to find this limit. I should be something way simpler.
$endgroup$
– The Cat
Jan 25 at 20:15
add a comment |
$begingroup$
I got stuck on the following limit:
$limlimits_{n to infty}5sum_{k=1}^n frac{sqrt{4n+5k}}{n sqrt{n}} $
The expected answer is $frac{38}{15}.$ Any hint is appreciated. Thanks
EDIT: The task is from this polish textbook (http://lucc.pl/inf/analiza_1/gewert_skoczylas__analiza_matematyczna_1__definicje_twierdzenia_wzory.pdf). It's on page 166 of the book (ex. 8.4.3 h). I want to compute this using the above fact. It's a book for first year students.
limits summation
asked Jan 25 at 20:09
The CatThe Cat
33815
$endgroup$
I got stuck on the following limit:
$limlimits_{n to infty}5sum_{k=1}^n frac{sqrt{4n+5k}}{n sqrt{n}} $
The expected answer is $frac{38}{15}.$ Any hint is appreciated. Thanks
EDIT: The task is from this polish textbook (http://lucc.pl/inf/analiza_1/gewert_skoczylas__analiza_matematyczna_1__definicje_twierdzenia_wzory.pdf). It's on page 166 of the book (ex. 8.4.3 h). I want to compute this using the above fact. It's a book for first year students.
limits summation
limits summation
asked Jan 25 at 20:09
The CatThe Cat
33815
asked Jan 25 at 20:09
The CatThe Cat
33815
edited Jan 25 at 22:32
The Cat
asked Jan 25 at 20:09
The CatThe Cat
33815
asked Jan 25 at 20:09
The CatThe Cat
33815
asked Jan 25 at 20:09
The CatThe Cat
33815
33815
$begingroup$
If you rearrange the terms a bit, this is a Riemann sum. So it converges to some integral. Can you take it from there?
$endgroup$
– Stefan Lafon
Jan 25 at 20:12
$begingroup$
Well this task here is from a prep-book to college. I've just started calculating definite integrals by definition, and in order to do the exercise i need to find this limit. I should be something way simpler.
$endgroup$
– The Cat
Jan 25 at 20:15
add a comment |
$begingroup$
If you rearrange the terms a bit, this is a Riemann sum. So it converges to some integral. Can you take it from there?
$endgroup$
– Stefan Lafon
Jan 25 at 20:12
$begingroup$
Well this task here is from a prep-book to college. I've just started calculating definite integrals by definition, and in order to do the exercise i need to find this limit. I should be something way simpler.
$endgroup$
– The Cat
Jan 25 at 20:15
$begingroup$
If you rearrange the terms a bit, this is a Riemann sum. So it converges to some integral. Can you take it from there?
$endgroup$
– Stefan Lafon
Jan 25 at 20:12
$begingroup$
If you rearrange the terms a bit, this is a Riemann sum. So it converges to some integral. Can you take it from there?
$endgroup$
– Stefan Lafon
Jan 25 at 20:12
$begingroup$
Well this task here is from a prep-book to college. I've just started calculating definite integrals by definition, and in order to do the exercise i need to find this limit. I should be something way simpler.
$endgroup$
– The Cat
Jan 25 at 20:15
$begingroup$
Well this task here is from a prep-book to college. I've just started calculating definite integrals by definition, and in order to do the exercise i need to find this limit. I should be something way simpler.
$endgroup$
– The Cat
Jan 25 at 20:15
add a comment |
1 Answer
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Hint (Riemman Sum-Integral) :
$$lim_{n to infty} sum_{k=1}^n frac{sqrt{4n+5k}}{nsqrt{n}} = lim_{n to infty} frac{1}{n} sum_{k=1}^n sqrt{4 + 5frac{k}{n}} = int_0^1 sqrt{4 + 5x} ;mathrm{d}x$$
answered Jan 25 at 20:12
RebellosRebellos
15.4k31250
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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$begingroup$
Hint (Riemman Sum-Integral) :
$$lim_{n to infty} sum_{k=1}^n frac{sqrt{4n+5k}}{nsqrt{n}} = lim_{n to infty} frac{1}{n} sum_{k=1}^n sqrt{4 + 5frac{k}{n}} = int_0^1 sqrt{4 + 5x} ;mathrm{d}x$$
answered Jan 25 at 20:12
RebellosRebellos
15.4k31250
$endgroup$
add a comment |
$begingroup$
Hint (Riemman Sum-Integral) :
$$lim_{n to infty} sum_{k=1}^n frac{sqrt{4n+5k}}{nsqrt{n}} = lim_{n to infty} frac{1}{n} sum_{k=1}^n sqrt{4 + 5frac{k}{n}} = int_0^1 sqrt{4 + 5x} ;mathrm{d}x$$
answered Jan 25 at 20:12
RebellosRebellos
15.4k31250
$endgroup$
add a comment |
$begingroup$
Hint (Riemman Sum-Integral) :
$$lim_{n to infty} sum_{k=1}^n frac{sqrt{4n+5k}}{nsqrt{n}} = lim_{n to infty} frac{1}{n} sum_{k=1}^n sqrt{4 + 5frac{k}{n}} = int_0^1 sqrt{4 + 5x} ;mathrm{d}x$$
answered Jan 25 at 20:12
RebellosRebellos
15.4k31250
$endgroup$
Hint (Riemman Sum-Integral) :
$$lim_{n to infty} sum_{k=1}^n frac{sqrt{4n+5k}}{nsqrt{n}} = lim_{n to infty} frac{1}{n} sum_{k=1}^n sqrt{4 + 5frac{k}{n}} = int_0^1 sqrt{4 + 5x} ;mathrm{d}x$$
answered Jan 25 at 20:12
RebellosRebellos
15.4k31250
answered Jan 25 at 20:12
RebellosRebellos
15.4k31250
answered Jan 25 at 20:12
RebellosRebellos
15.4k31250
answered Jan 25 at 20:12
RebellosRebellos
15.4k31250
15.4k31250
add a comment |
add a comment |
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$begingroup$
If you rearrange the terms a bit, this is a Riemann sum. So it converges to some integral. Can you take it from there?
$endgroup$
– Stefan Lafon
Jan 25 at 20:12
$begingroup$
Well this task here is from a prep-book to college. I've just started calculating definite integrals by definition, and in order to do the exercise i need to find this limit. I should be something way simpler.
$endgroup$
– The Cat
Jan 25 at 20:15