Let $a_n = intlimits_{0}^{n} e^{-x^4} dx$. Does ${ a_n }_{n rightarrow infty}$ converge?
$begingroup$
Let $a_n = intlimits_{0}^{n} e^{-x^4} dx$. Does ${ a_n }_{n rightarrow infty}$ converge?
${ a_n } ={ intlimits_{0}^{1} e^{-x^4} dx, intlimits_{0}^{2} e^{-x^4} dx, ..., intlimits_{0}^{infty} e^{-x^4} dx }$
So we need need only to check that the definite integral
$intlimits_{0}^{infty} e^{-x^4} dx$ converges
By using Wolfram Alpha,
$intlimits_{0}^{infty} e^{-x^4} dx } = Gamma left( frac54 right) approx 0.906402$
where $Gamma$ is the Gamma function
Therefore ${ a_n }$ converges, to $Gamma left( frac54 right)$.
But what if I can't use Wolfram Alpha? How can I solve for this integration by hand? Sorry if this makes me look stupid, I don't recall learning any techniques from Calculus that can help me solve this integration.
Thanks in advance!
calculus integration sequences-and-series gamma-function
$endgroup$
add a comment |
$begingroup$
Let $a_n = intlimits_{0}^{n} e^{-x^4} dx$. Does ${ a_n }_{n rightarrow infty}$ converge?
${ a_n } ={ intlimits_{0}^{1} e^{-x^4} dx, intlimits_{0}^{2} e^{-x^4} dx, ..., intlimits_{0}^{infty} e^{-x^4} dx }$
So we need need only to check that the definite integral
$intlimits_{0}^{infty} e^{-x^4} dx$ converges
By using Wolfram Alpha,
$intlimits_{0}^{infty} e^{-x^4} dx } = Gamma left( frac54 right) approx 0.906402$
where $Gamma$ is the Gamma function
Therefore ${ a_n }$ converges, to $Gamma left( frac54 right)$.
But what if I can't use Wolfram Alpha? How can I solve for this integration by hand? Sorry if this makes me look stupid, I don't recall learning any techniques from Calculus that can help me solve this integration.
Thanks in advance!
calculus integration sequences-and-series gamma-function
$endgroup$
1
$begingroup$
Exponential are 'always' quite strong.
$endgroup$
– Felix Marin
Oct 31 '13 at 7:47
1
$begingroup$
Generally speaking, $$n!=Gleft(frac1nright)qquad where qquad G(n)=int_0^infty e^{-x^n}dxqquadforall ngeqslant0$$
$endgroup$
– Lucian
Oct 31 '13 at 7:47
add a comment |
$begingroup$
Let $a_n = intlimits_{0}^{n} e^{-x^4} dx$. Does ${ a_n }_{n rightarrow infty}$ converge?
${ a_n } ={ intlimits_{0}^{1} e^{-x^4} dx, intlimits_{0}^{2} e^{-x^4} dx, ..., intlimits_{0}^{infty} e^{-x^4} dx }$
So we need need only to check that the definite integral
$intlimits_{0}^{infty} e^{-x^4} dx$ converges
By using Wolfram Alpha,
$intlimits_{0}^{infty} e^{-x^4} dx } = Gamma left( frac54 right) approx 0.906402$
where $Gamma$ is the Gamma function
Therefore ${ a_n }$ converges, to $Gamma left( frac54 right)$.
But what if I can't use Wolfram Alpha? How can I solve for this integration by hand? Sorry if this makes me look stupid, I don't recall learning any techniques from Calculus that can help me solve this integration.
Thanks in advance!
calculus integration sequences-and-series gamma-function
$endgroup$
Let $a_n = intlimits_{0}^{n} e^{-x^4} dx$. Does ${ a_n }_{n rightarrow infty}$ converge?
${ a_n } ={ intlimits_{0}^{1} e^{-x^4} dx, intlimits_{0}^{2} e^{-x^4} dx, ..., intlimits_{0}^{infty} e^{-x^4} dx }$
So we need need only to check that the definite integral
$intlimits_{0}^{infty} e^{-x^4} dx$ converges
By using Wolfram Alpha,
$intlimits_{0}^{infty} e^{-x^4} dx } = Gamma left( frac54 right) approx 0.906402$
where $Gamma$ is the Gamma function
Therefore ${ a_n }$ converges, to $Gamma left( frac54 right)$.
But what if I can't use Wolfram Alpha? How can I solve for this integration by hand? Sorry if this makes me look stupid, I don't recall learning any techniques from Calculus that can help me solve this integration.
Thanks in advance!
calculus integration sequences-and-series gamma-function
calculus integration sequences-and-series gamma-function
asked Oct 31 '13 at 7:33
PandaManPandaMan
1,19911333
1,19911333
1
$begingroup$
Exponential are 'always' quite strong.
$endgroup$
– Felix Marin
Oct 31 '13 at 7:47
1
$begingroup$
Generally speaking, $$n!=Gleft(frac1nright)qquad where qquad G(n)=int_0^infty e^{-x^n}dxqquadforall ngeqslant0$$
$endgroup$
– Lucian
Oct 31 '13 at 7:47
add a comment |
1
$begingroup$
Exponential are 'always' quite strong.
$endgroup$
– Felix Marin
Oct 31 '13 at 7:47
1
$begingroup$
Generally speaking, $$n!=Gleft(frac1nright)qquad where qquad G(n)=int_0^infty e^{-x^n}dxqquadforall ngeqslant0$$
$endgroup$
– Lucian
Oct 31 '13 at 7:47
1
1
$begingroup$
Exponential are 'always' quite strong.
$endgroup$
– Felix Marin
Oct 31 '13 at 7:47
$begingroup$
Exponential are 'always' quite strong.
$endgroup$
– Felix Marin
Oct 31 '13 at 7:47
1
1
$begingroup$
Generally speaking, $$n!=Gleft(frac1nright)qquad where qquad G(n)=int_0^infty e^{-x^n}dxqquadforall ngeqslant0$$
$endgroup$
– Lucian
Oct 31 '13 at 7:47
$begingroup$
Generally speaking, $$n!=Gleft(frac1nright)qquad where qquad G(n)=int_0^infty e^{-x^n}dxqquadforall ngeqslant0$$
$endgroup$
– Lucian
Oct 31 '13 at 7:47
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
We do not need an explicit expression to show that an improper integral converges.
The sequence $(a_n)$ is obviously increasing. It is bounded above by $int_0^1 e^{-x^4},dx+int_1^infty e^{-x},dx$. To be explicit, the first integral is less than $1$, and the second is $e^{-1}$, so the sequence $(a_n)$ is bounded above by $1+e^{-1}$.
Any increasing sequence which is bounded above converges.
$endgroup$
$begingroup$
You are welcome.
$endgroup$
– André Nicolas
Nov 19 '13 at 3:07
add a comment |
$begingroup$
Recall the Taylor series for $e^x$ at $x = 0$ is given by:
begin{equation}
e^{x} = sum_{n = 0}^{infty} frac{x^n}{n!}
end{equation}
Thus,
begin{equation}
e^{-x^4} = sum_{n = 0}^{infty} frac{(-1)^n x^{4n}}{n!}
end{equation}
Thus,
begin{align}
int_{0}^{t} e^{-x^4} :dx &= sum_{n = 0}^{infty} frac{(-1)^n }{n!}int_{0}^{t}x^{4n}:dx = sum_{n = 0}^{infty} frac{(-1)^n }{n!} left[frac{x^{4n + 1}}{4n + 1} right]_{0}^{t} \
&= sum_{n = 0}^{infty} frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1} = sum_{n = 0}^{infty} b_n(t)
end{align}
We now apply the Ratio Test:
begin{align}
R &= lim_{n rightarrow infty} left|frac{b_{n + 1}}{b_{n}}right| = lim_{n rightarrow infty} left|frac{frac{(-1)^{n+1} }{left(n + 1right)!} frac{t^{4left(n + 1right) + 1}}{4left(n + 1right) + 1}}{frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1}}right| \
&= lim_{n rightarrow infty} left| frac{1}{n + 1} cdot frac{4n + 1}{4n + 5}cdot t^4right|
end{align}
Which for any finite $t$ becomes $R = 0$ and thus, by the Ratio test the series (and by extension) integral converge for all finite real $t$.
$endgroup$
add a comment |
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2 Answers
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active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
We do not need an explicit expression to show that an improper integral converges.
The sequence $(a_n)$ is obviously increasing. It is bounded above by $int_0^1 e^{-x^4},dx+int_1^infty e^{-x},dx$. To be explicit, the first integral is less than $1$, and the second is $e^{-1}$, so the sequence $(a_n)$ is bounded above by $1+e^{-1}$.
Any increasing sequence which is bounded above converges.
$endgroup$
$begingroup$
You are welcome.
$endgroup$
– André Nicolas
Nov 19 '13 at 3:07
add a comment |
$begingroup$
We do not need an explicit expression to show that an improper integral converges.
The sequence $(a_n)$ is obviously increasing. It is bounded above by $int_0^1 e^{-x^4},dx+int_1^infty e^{-x},dx$. To be explicit, the first integral is less than $1$, and the second is $e^{-1}$, so the sequence $(a_n)$ is bounded above by $1+e^{-1}$.
Any increasing sequence which is bounded above converges.
$endgroup$
$begingroup$
You are welcome.
$endgroup$
– André Nicolas
Nov 19 '13 at 3:07
add a comment |
$begingroup$
We do not need an explicit expression to show that an improper integral converges.
The sequence $(a_n)$ is obviously increasing. It is bounded above by $int_0^1 e^{-x^4},dx+int_1^infty e^{-x},dx$. To be explicit, the first integral is less than $1$, and the second is $e^{-1}$, so the sequence $(a_n)$ is bounded above by $1+e^{-1}$.
Any increasing sequence which is bounded above converges.
$endgroup$
We do not need an explicit expression to show that an improper integral converges.
The sequence $(a_n)$ is obviously increasing. It is bounded above by $int_0^1 e^{-x^4},dx+int_1^infty e^{-x},dx$. To be explicit, the first integral is less than $1$, and the second is $e^{-1}$, so the sequence $(a_n)$ is bounded above by $1+e^{-1}$.
Any increasing sequence which is bounded above converges.
edited Oct 31 '13 at 7:58
answered Oct 31 '13 at 7:40
André NicolasAndré Nicolas
455k36432819
455k36432819
$begingroup$
You are welcome.
$endgroup$
– André Nicolas
Nov 19 '13 at 3:07
add a comment |
$begingroup$
You are welcome.
$endgroup$
– André Nicolas
Nov 19 '13 at 3:07
$begingroup$
You are welcome.
$endgroup$
– André Nicolas
Nov 19 '13 at 3:07
$begingroup$
You are welcome.
$endgroup$
– André Nicolas
Nov 19 '13 at 3:07
add a comment |
$begingroup$
Recall the Taylor series for $e^x$ at $x = 0$ is given by:
begin{equation}
e^{x} = sum_{n = 0}^{infty} frac{x^n}{n!}
end{equation}
Thus,
begin{equation}
e^{-x^4} = sum_{n = 0}^{infty} frac{(-1)^n x^{4n}}{n!}
end{equation}
Thus,
begin{align}
int_{0}^{t} e^{-x^4} :dx &= sum_{n = 0}^{infty} frac{(-1)^n }{n!}int_{0}^{t}x^{4n}:dx = sum_{n = 0}^{infty} frac{(-1)^n }{n!} left[frac{x^{4n + 1}}{4n + 1} right]_{0}^{t} \
&= sum_{n = 0}^{infty} frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1} = sum_{n = 0}^{infty} b_n(t)
end{align}
We now apply the Ratio Test:
begin{align}
R &= lim_{n rightarrow infty} left|frac{b_{n + 1}}{b_{n}}right| = lim_{n rightarrow infty} left|frac{frac{(-1)^{n+1} }{left(n + 1right)!} frac{t^{4left(n + 1right) + 1}}{4left(n + 1right) + 1}}{frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1}}right| \
&= lim_{n rightarrow infty} left| frac{1}{n + 1} cdot frac{4n + 1}{4n + 5}cdot t^4right|
end{align}
Which for any finite $t$ becomes $R = 0$ and thus, by the Ratio test the series (and by extension) integral converge for all finite real $t$.
$endgroup$
add a comment |
$begingroup$
Recall the Taylor series for $e^x$ at $x = 0$ is given by:
begin{equation}
e^{x} = sum_{n = 0}^{infty} frac{x^n}{n!}
end{equation}
Thus,
begin{equation}
e^{-x^4} = sum_{n = 0}^{infty} frac{(-1)^n x^{4n}}{n!}
end{equation}
Thus,
begin{align}
int_{0}^{t} e^{-x^4} :dx &= sum_{n = 0}^{infty} frac{(-1)^n }{n!}int_{0}^{t}x^{4n}:dx = sum_{n = 0}^{infty} frac{(-1)^n }{n!} left[frac{x^{4n + 1}}{4n + 1} right]_{0}^{t} \
&= sum_{n = 0}^{infty} frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1} = sum_{n = 0}^{infty} b_n(t)
end{align}
We now apply the Ratio Test:
begin{align}
R &= lim_{n rightarrow infty} left|frac{b_{n + 1}}{b_{n}}right| = lim_{n rightarrow infty} left|frac{frac{(-1)^{n+1} }{left(n + 1right)!} frac{t^{4left(n + 1right) + 1}}{4left(n + 1right) + 1}}{frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1}}right| \
&= lim_{n rightarrow infty} left| frac{1}{n + 1} cdot frac{4n + 1}{4n + 5}cdot t^4right|
end{align}
Which for any finite $t$ becomes $R = 0$ and thus, by the Ratio test the series (and by extension) integral converge for all finite real $t$.
$endgroup$
add a comment |
$begingroup$
Recall the Taylor series for $e^x$ at $x = 0$ is given by:
begin{equation}
e^{x} = sum_{n = 0}^{infty} frac{x^n}{n!}
end{equation}
Thus,
begin{equation}
e^{-x^4} = sum_{n = 0}^{infty} frac{(-1)^n x^{4n}}{n!}
end{equation}
Thus,
begin{align}
int_{0}^{t} e^{-x^4} :dx &= sum_{n = 0}^{infty} frac{(-1)^n }{n!}int_{0}^{t}x^{4n}:dx = sum_{n = 0}^{infty} frac{(-1)^n }{n!} left[frac{x^{4n + 1}}{4n + 1} right]_{0}^{t} \
&= sum_{n = 0}^{infty} frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1} = sum_{n = 0}^{infty} b_n(t)
end{align}
We now apply the Ratio Test:
begin{align}
R &= lim_{n rightarrow infty} left|frac{b_{n + 1}}{b_{n}}right| = lim_{n rightarrow infty} left|frac{frac{(-1)^{n+1} }{left(n + 1right)!} frac{t^{4left(n + 1right) + 1}}{4left(n + 1right) + 1}}{frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1}}right| \
&= lim_{n rightarrow infty} left| frac{1}{n + 1} cdot frac{4n + 1}{4n + 5}cdot t^4right|
end{align}
Which for any finite $t$ becomes $R = 0$ and thus, by the Ratio test the series (and by extension) integral converge for all finite real $t$.
$endgroup$
Recall the Taylor series for $e^x$ at $x = 0$ is given by:
begin{equation}
e^{x} = sum_{n = 0}^{infty} frac{x^n}{n!}
end{equation}
Thus,
begin{equation}
e^{-x^4} = sum_{n = 0}^{infty} frac{(-1)^n x^{4n}}{n!}
end{equation}
Thus,
begin{align}
int_{0}^{t} e^{-x^4} :dx &= sum_{n = 0}^{infty} frac{(-1)^n }{n!}int_{0}^{t}x^{4n}:dx = sum_{n = 0}^{infty} frac{(-1)^n }{n!} left[frac{x^{4n + 1}}{4n + 1} right]_{0}^{t} \
&= sum_{n = 0}^{infty} frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1} = sum_{n = 0}^{infty} b_n(t)
end{align}
We now apply the Ratio Test:
begin{align}
R &= lim_{n rightarrow infty} left|frac{b_{n + 1}}{b_{n}}right| = lim_{n rightarrow infty} left|frac{frac{(-1)^{n+1} }{left(n + 1right)!} frac{t^{4left(n + 1right) + 1}}{4left(n + 1right) + 1}}{frac{(-1)^n }{n!} frac{t^{4n + 1}}{4n + 1}}right| \
&= lim_{n rightarrow infty} left| frac{1}{n + 1} cdot frac{4n + 1}{4n + 5}cdot t^4right|
end{align}
Which for any finite $t$ becomes $R = 0$ and thus, by the Ratio test the series (and by extension) integral converge for all finite real $t$.
answered Dec 19 '18 at 2:28
user150203
add a comment |
add a comment |
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$begingroup$
Exponential are 'always' quite strong.
$endgroup$
– Felix Marin
Oct 31 '13 at 7:47
1
$begingroup$
Generally speaking, $$n!=Gleft(frac1nright)qquad where qquad G(n)=int_0^infty e^{-x^n}dxqquadforall ngeqslant0$$
$endgroup$
– Lucian
Oct 31 '13 at 7:47