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How to turn recursion sum formula into an integral?
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I have a formula that looks a lot like integral:
$$
m(t) = lim_Delta t to 0 sum_i in Delta t, 2Delta t, ... ,t ( m (i - Delta t) + v(i)) , Delta t
$$
where $v(t) = textconst$, $m(0)$ is given.
Yet I wonder - how to turn it into an integral and is it possible when to calculate $m(t)$ I have to obtain $mleft(t- Delta tright)$ ?
integration definite-integrals improper-integrals
edited Jul 23 at 11:54
Sangchul Lee
85.5k12155253
asked Jul 22 at 11:58
DuckQueen
1377
add a comment |Â
up vote
0
down vote
favorite
I have a formula that looks a lot like integral:
$$
m(t) = lim_Delta t to 0 sum_i in Delta t, 2Delta t, ... ,t ( m (i - Delta t) + v(i)) , Delta t
$$
where $v(t) = textconst$, $m(0)$ is given.
Yet I wonder - how to turn it into an integral and is it possible when to calculate $m(t)$ I have to obtain $mleft(t- Delta tright)$ ?
integration definite-integrals improper-integrals
edited Jul 23 at 11:54
Sangchul Lee
85.5k12155253
asked Jul 22 at 11:58
DuckQueen
1377
Without any context we can't help you. What is "Pr(t)"? What is "n(t)"? What is "v(t)"? Where is "i" used in the sum?
– Somos
Jul 22 at 13:43
They are functions that depend only on time
– DuckQueen
Jul 22 at 22:01
simplified a bit
– DuckQueen
Jul 23 at 10:35
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a formula that looks a lot like integral:
$$
m(t) = lim_Delta t to 0 sum_i in Delta t, 2Delta t, ... ,t ( m (i - Delta t) + v(i)) , Delta t
$$
where $v(t) = textconst$, $m(0)$ is given.
Yet I wonder - how to turn it into an integral and is it possible when to calculate $m(t)$ I have to obtain $mleft(t- Delta tright)$ ?
integration definite-integrals improper-integrals
edited Jul 23 at 11:54
Sangchul Lee
85.5k12155253
asked Jul 22 at 11:58
DuckQueen
1377
I have a formula that looks a lot like integral:
$$
m(t) = lim_Delta t to 0 sum_i in Delta t, 2Delta t, ... ,t ( m (i - Delta t) + v(i)) , Delta t
$$
where $v(t) = textconst$, $m(0)$ is given.
Yet I wonder - how to turn it into an integral and is it possible when to calculate $m(t)$ I have to obtain $mleft(t- Delta tright)$ ?
integration definite-integrals improper-integrals
edited Jul 23 at 11:54
Sangchul Lee
85.5k12155253
asked Jul 22 at 11:58
DuckQueen
1377
edited Jul 23 at 11:54
Sangchul Lee
85.5k12155253
edited Jul 23 at 11:54
Sangchul Lee
85.5k12155253
edited Jul 23 at 11:54
Sangchul Lee
85.5k12155253
85.5k12155253
asked Jul 22 at 11:58
DuckQueen
1377
asked Jul 22 at 11:58
DuckQueen
1377
asked Jul 22 at 11:58
DuckQueen
1377
1377
Without any context we can't help you. What is "Pr(t)"? What is "n(t)"? What is "v(t)"? Where is "i" used in the sum?
– Somos
Jul 22 at 13:43
They are functions that depend only on time
– DuckQueen
Jul 22 at 22:01
simplified a bit
– DuckQueen
Jul 23 at 10:35
add a comment |Â
Without any context we can't help you. What is "Pr(t)"? What is "n(t)"? What is "v(t)"? Where is "i" used in the sum?
– Somos
Jul 22 at 13:43
They are functions that depend only on time
– DuckQueen
Jul 22 at 22:01
simplified a bit
– DuckQueen
Jul 23 at 10:35
Without any context we can't help you. What is "Pr(t)"? What is "n(t)"? What is "v(t)"? Where is "i" used in the sum?
– Somos
Jul 22 at 13:43
Without any context we can't help you. What is "Pr(t)"? What is "n(t)"? What is "v(t)"? Where is "i" used in the sum?
– Somos
Jul 22 at 13:43
They are functions that depend only on time
– DuckQueen
Jul 22 at 22:01
They are functions that depend only on time
– DuckQueen
Jul 22 at 22:01
simplified a bit
– DuckQueen
Jul 23 at 10:35
simplified a bit
– DuckQueen
Jul 23 at 10:35
add a comment |Â
1 Answer
1
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oldest
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0
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The right side of your formula is a limit. If that limit exists, then it is an integral as you suggested. $, m(t) = int_0^t m(x) + v(x), dx. ,$ Your question about $, m(t-Delta t) ,$ turns out not to be a problem because in the limit that is the same as $, m(t). ,$ Taking the derivative of both sides of the equation, $, m'(t) = m(t) + v(t) ,$ which is a simple differential equation with initial value $, m(0)=0. ,$
answered Jul 23 at 10:45
Somos
11.5k1933
but how to show that $m(x)$ has $x$ that is less than $v(x)$ and thus $m(x)$ has to be acquired on previous integration step?
– DuckQueen
Jul 23 at 11:04
@DuckQueen What does "m(x) has x that is less than v(x)" mean? What previous integration step? Your comment does not make sense to me. Put your sensible comment in the question itself if it is important to you.
– Somos
Jul 29 at 0:40
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The right side of your formula is a limit. If that limit exists, then it is an integral as you suggested. $, m(t) = int_0^t m(x) + v(x), dx. ,$ Your question about $, m(t-Delta t) ,$ turns out not to be a problem because in the limit that is the same as $, m(t). ,$ Taking the derivative of both sides of the equation, $, m'(t) = m(t) + v(t) ,$ which is a simple differential equation with initial value $, m(0)=0. ,$
answered Jul 23 at 10:45
Somos
11.5k1933
but how to show that $m(x)$ has $x$ that is less than $v(x)$ and thus $m(x)$ has to be acquired on previous integration step?
– DuckQueen
Jul 23 at 11:04
@DuckQueen What does "m(x) has x that is less than v(x)" mean? What previous integration step? Your comment does not make sense to me. Put your sensible comment in the question itself if it is important to you.
– Somos
Jul 29 at 0:40
add a comment |Â
up vote
0
down vote
The right side of your formula is a limit. If that limit exists, then it is an integral as you suggested. $, m(t) = int_0^t m(x) + v(x), dx. ,$ Your question about $, m(t-Delta t) ,$ turns out not to be a problem because in the limit that is the same as $, m(t). ,$ Taking the derivative of both sides of the equation, $, m'(t) = m(t) + v(t) ,$ which is a simple differential equation with initial value $, m(0)=0. ,$
answered Jul 23 at 10:45
Somos
11.5k1933
but how to show that $m(x)$ has $x$ that is less than $v(x)$ and thus $m(x)$ has to be acquired on previous integration step?
– DuckQueen
Jul 23 at 11:04
@DuckQueen What does "m(x) has x that is less than v(x)" mean? What previous integration step? Your comment does not make sense to me. Put your sensible comment in the question itself if it is important to you.
– Somos
Jul 29 at 0:40
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The right side of your formula is a limit. If that limit exists, then it is an integral as you suggested. $, m(t) = int_0^t m(x) + v(x), dx. ,$ Your question about $, m(t-Delta t) ,$ turns out not to be a problem because in the limit that is the same as $, m(t). ,$ Taking the derivative of both sides of the equation, $, m'(t) = m(t) + v(t) ,$ which is a simple differential equation with initial value $, m(0)=0. ,$
answered Jul 23 at 10:45
Somos
11.5k1933
The right side of your formula is a limit. If that limit exists, then it is an integral as you suggested. $, m(t) = int_0^t m(x) + v(x), dx. ,$ Your question about $, m(t-Delta t) ,$ turns out not to be a problem because in the limit that is the same as $, m(t). ,$ Taking the derivative of both sides of the equation, $, m'(t) = m(t) + v(t) ,$ which is a simple differential equation with initial value $, m(0)=0. ,$
answered Jul 23 at 10:45
Somos
11.5k1933
edited Jul 29 at 0:37
answered Jul 23 at 10:45
Somos
11.5k1933
answered Jul 23 at 10:45
Somos
11.5k1933
answered Jul 23 at 10:45
Somos
11.5k1933
11.5k1933
but how to show that $m(x)$ has $x$ that is less than $v(x)$ and thus $m(x)$ has to be acquired on previous integration step?
– DuckQueen
Jul 23 at 11:04
@DuckQueen What does "m(x) has x that is less than v(x)" mean? What previous integration step? Your comment does not make sense to me. Put your sensible comment in the question itself if it is important to you.
– Somos
Jul 29 at 0:40
add a comment |Â
but how to show that $m(x)$ has $x$ that is less than $v(x)$ and thus $m(x)$ has to be acquired on previous integration step?
– DuckQueen
Jul 23 at 11:04
@DuckQueen What does "m(x) has x that is less than v(x)" mean? What previous integration step? Your comment does not make sense to me. Put your sensible comment in the question itself if it is important to you.
– Somos
Jul 29 at 0:40
but how to show that $m(x)$ has $x$ that is less than $v(x)$ and thus $m(x)$ has to be acquired on previous integration step?
– DuckQueen
Jul 23 at 11:04
but how to show that $m(x)$ has $x$ that is less than $v(x)$ and thus $m(x)$ has to be acquired on previous integration step?
– DuckQueen
Jul 23 at 11:04
@DuckQueen What does "m(x) has x that is less than v(x)" mean? What previous integration step? Your comment does not make sense to me. Put your sensible comment in the question itself if it is important to you.
– Somos
Jul 29 at 0:40
@DuckQueen What does "m(x) has x that is less than v(x)" mean? What previous integration step? Your comment does not make sense to me. Put your sensible comment in the question itself if it is important to you.
– Somos
Jul 29 at 0:40
add a comment |Â
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Without any context we can't help you. What is "Pr(t)"? What is "n(t)"? What is "v(t)"? Where is "i" used in the sum?
– Somos
Jul 22 at 13:43
They are functions that depend only on time
– DuckQueen
Jul 22 at 22:01
simplified a bit
– DuckQueen
Jul 23 at 10:35