Calculating the limit $lim_{ntoinfty}[sum_{k=0}^{n-1}(-1)^kfrac{1}{n-k}]$
$begingroup$
So, I have got the series: $$I_n = int_0^1frac{x^n}{1+x}dx$$ and my task is to find the limit
$$lim_{ntoinfty}(I_n)$$
I have added 1 and subtracted 1 from the numerator and factorised $x^n - 1$ as $(x+1)(x^{n-1}-x^{n-2}+x^{n-3} - ...+(-1)^n)$. Then i did the reduction, and I'm left with calculating the limit:
$$lim_{ntoinfty}[frac{1}{n}-frac{1}{n-1}+frac{1}{n-2} -...+(-1)^{n-1}frac{1}{2} + (-1)^n] = lim_{ntoinfty}[sum_{k=0}^{n-1}(-1)^kfrac{1}{n-k}]$$
I know I can calculate the fraction as a Riemann Sum using the integral of the function $frac{1}{1-x}$, however I don't know how to deal with the changing signs. How should i approach this? Is there some sort of rule or common practice to deal with alternating signs?
sequences-and-series limits
$endgroup$
add a comment |
$begingroup$
So, I have got the series: $$I_n = int_0^1frac{x^n}{1+x}dx$$ and my task is to find the limit
$$lim_{ntoinfty}(I_n)$$
I have added 1 and subtracted 1 from the numerator and factorised $x^n - 1$ as $(x+1)(x^{n-1}-x^{n-2}+x^{n-3} - ...+(-1)^n)$. Then i did the reduction, and I'm left with calculating the limit:
$$lim_{ntoinfty}[frac{1}{n}-frac{1}{n-1}+frac{1}{n-2} -...+(-1)^{n-1}frac{1}{2} + (-1)^n] = lim_{ntoinfty}[sum_{k=0}^{n-1}(-1)^kfrac{1}{n-k}]$$
I know I can calculate the fraction as a Riemann Sum using the integral of the function $frac{1}{1-x}$, however I don't know how to deal with the changing signs. How should i approach this? Is there some sort of rule or common practice to deal with alternating signs?
sequences-and-series limits
$endgroup$
add a comment |
$begingroup$
So, I have got the series: $$I_n = int_0^1frac{x^n}{1+x}dx$$ and my task is to find the limit
$$lim_{ntoinfty}(I_n)$$
I have added 1 and subtracted 1 from the numerator and factorised $x^n - 1$ as $(x+1)(x^{n-1}-x^{n-2}+x^{n-3} - ...+(-1)^n)$. Then i did the reduction, and I'm left with calculating the limit:
$$lim_{ntoinfty}[frac{1}{n}-frac{1}{n-1}+frac{1}{n-2} -...+(-1)^{n-1}frac{1}{2} + (-1)^n] = lim_{ntoinfty}[sum_{k=0}^{n-1}(-1)^kfrac{1}{n-k}]$$
I know I can calculate the fraction as a Riemann Sum using the integral of the function $frac{1}{1-x}$, however I don't know how to deal with the changing signs. How should i approach this? Is there some sort of rule or common practice to deal with alternating signs?
sequences-and-series limits
$endgroup$
So, I have got the series: $$I_n = int_0^1frac{x^n}{1+x}dx$$ and my task is to find the limit
$$lim_{ntoinfty}(I_n)$$
I have added 1 and subtracted 1 from the numerator and factorised $x^n - 1$ as $(x+1)(x^{n-1}-x^{n-2}+x^{n-3} - ...+(-1)^n)$. Then i did the reduction, and I'm left with calculating the limit:
$$lim_{ntoinfty}[frac{1}{n}-frac{1}{n-1}+frac{1}{n-2} -...+(-1)^{n-1}frac{1}{2} + (-1)^n] = lim_{ntoinfty}[sum_{k=0}^{n-1}(-1)^kfrac{1}{n-k}]$$
I know I can calculate the fraction as a Riemann Sum using the integral of the function $frac{1}{1-x}$, however I don't know how to deal with the changing signs. How should i approach this? Is there some sort of rule or common practice to deal with alternating signs?
sequences-and-series limits
sequences-and-series limits
edited Jan 25 at 8:36
Constantin
asked Jan 24 at 21:20
ConstantinConstantin
335
335
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3 Answers
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$begingroup$
$$I_nleqint_0^1x^ndx=frac 1{n+1}$$ so $$lim_{nrightarrow +infty}I_n=0$$
$endgroup$
$begingroup$
Just to make sure: you are basing this on the squeeze theorem? 0 <= In <= 1/(n+1) therefore 0 <= lim(In) <= 0 when n goes to infinity?
$endgroup$
– Constantin
Jan 25 at 15:27
$begingroup$
That's right. You got it.
$endgroup$
– Stefan Lafon
Jan 25 at 15:34
add a comment |
$begingroup$
Note that for all $n$ we have :
$$forall xin [0,1], frac{x^n}{1+x} leq 1$$
A constant function is clearly integrable, hence we can apply the dominated convergence theorem to get :
$$lim_{n to infty}int_0^1 frac{x^n}{1+x} mathrm{d}x = int_0^1, lim_{n to infty} frac{x^n}{1+x} mathrm{d}x = 0$$
$endgroup$
add a comment |
$begingroup$
Proceeding as in the OP, note that we have two cases
$$frac{1+x^n}{1+x}=begin{cases}sum_{k=1}^{n} (-1)^{k-1}x^{k-1}&,n,,text{odd}\\
frac2{1+x}-sum_{k=1}^{n}(-1)^{k-1}x^{k-1}&,n,,text{even}end{cases}$$
Integrating over $[0,1]$, letting $nto infty$, and using the Taylor series representation for $log(1+x)$ for $x=1$ reveals
$$lim_{ntoinfty}int_0^1 frac{1+x^n}{1+x},dx=log(2) $$
Hence, we have
$$lim_{ntoinfty}int_0^1 frac{x^n}{1+x},dx=log(2)-log(2)=0$$
Rather than proceed as in the OP, we write $frac1{1+x}=sum_{m=0}^{infty}(-x)^m$. Then, we see that
$$begin{align}
int_0^1 frac{x^n}{1+x} ,dx&=sum_{m=0}^{infty} (-1)^m int_0^1 x^{n+m},dx\\
&=sum_{m=n+1}^{infty}(-1)^{m-n-1} frac1{m}tag1
end{align}$$
As the series $sum_{m=1}^{infty}frac{(-1)^{m-n-1}}{m}$ converges, the series on the right hand side of $(1)$ approaches $0$ as $nto infty$.
$endgroup$
$begingroup$
Please let me know how I can improve my answer. I really want to give you the best answer I can. And Happy New Year! ;-)
$endgroup$
– Mark Viola
Jan 30 at 5:04
add a comment |
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3 Answers
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3 Answers
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$begingroup$
$$I_nleqint_0^1x^ndx=frac 1{n+1}$$ so $$lim_{nrightarrow +infty}I_n=0$$
$endgroup$
$begingroup$
Just to make sure: you are basing this on the squeeze theorem? 0 <= In <= 1/(n+1) therefore 0 <= lim(In) <= 0 when n goes to infinity?
$endgroup$
– Constantin
Jan 25 at 15:27
$begingroup$
That's right. You got it.
$endgroup$
– Stefan Lafon
Jan 25 at 15:34
add a comment |
$begingroup$
$$I_nleqint_0^1x^ndx=frac 1{n+1}$$ so $$lim_{nrightarrow +infty}I_n=0$$
$endgroup$
$begingroup$
Just to make sure: you are basing this on the squeeze theorem? 0 <= In <= 1/(n+1) therefore 0 <= lim(In) <= 0 when n goes to infinity?
$endgroup$
– Constantin
Jan 25 at 15:27
$begingroup$
That's right. You got it.
$endgroup$
– Stefan Lafon
Jan 25 at 15:34
add a comment |
$begingroup$
$$I_nleqint_0^1x^ndx=frac 1{n+1}$$ so $$lim_{nrightarrow +infty}I_n=0$$
$endgroup$
$$I_nleqint_0^1x^ndx=frac 1{n+1}$$ so $$lim_{nrightarrow +infty}I_n=0$$
answered Jan 24 at 21:29
Stefan LafonStefan Lafon
2,83519
2,83519
$begingroup$
Just to make sure: you are basing this on the squeeze theorem? 0 <= In <= 1/(n+1) therefore 0 <= lim(In) <= 0 when n goes to infinity?
$endgroup$
– Constantin
Jan 25 at 15:27
$begingroup$
That's right. You got it.
$endgroup$
– Stefan Lafon
Jan 25 at 15:34
add a comment |
$begingroup$
Just to make sure: you are basing this on the squeeze theorem? 0 <= In <= 1/(n+1) therefore 0 <= lim(In) <= 0 when n goes to infinity?
$endgroup$
– Constantin
Jan 25 at 15:27
$begingroup$
That's right. You got it.
$endgroup$
– Stefan Lafon
Jan 25 at 15:34
$begingroup$
Just to make sure: you are basing this on the squeeze theorem? 0 <= In <= 1/(n+1) therefore 0 <= lim(In) <= 0 when n goes to infinity?
$endgroup$
– Constantin
Jan 25 at 15:27
$begingroup$
Just to make sure: you are basing this on the squeeze theorem? 0 <= In <= 1/(n+1) therefore 0 <= lim(In) <= 0 when n goes to infinity?
$endgroup$
– Constantin
Jan 25 at 15:27
$begingroup$
That's right. You got it.
$endgroup$
– Stefan Lafon
Jan 25 at 15:34
$begingroup$
That's right. You got it.
$endgroup$
– Stefan Lafon
Jan 25 at 15:34
add a comment |
$begingroup$
Note that for all $n$ we have :
$$forall xin [0,1], frac{x^n}{1+x} leq 1$$
A constant function is clearly integrable, hence we can apply the dominated convergence theorem to get :
$$lim_{n to infty}int_0^1 frac{x^n}{1+x} mathrm{d}x = int_0^1, lim_{n to infty} frac{x^n}{1+x} mathrm{d}x = 0$$
$endgroup$
add a comment |
$begingroup$
Note that for all $n$ we have :
$$forall xin [0,1], frac{x^n}{1+x} leq 1$$
A constant function is clearly integrable, hence we can apply the dominated convergence theorem to get :
$$lim_{n to infty}int_0^1 frac{x^n}{1+x} mathrm{d}x = int_0^1, lim_{n to infty} frac{x^n}{1+x} mathrm{d}x = 0$$
$endgroup$
add a comment |
$begingroup$
Note that for all $n$ we have :
$$forall xin [0,1], frac{x^n}{1+x} leq 1$$
A constant function is clearly integrable, hence we can apply the dominated convergence theorem to get :
$$lim_{n to infty}int_0^1 frac{x^n}{1+x} mathrm{d}x = int_0^1, lim_{n to infty} frac{x^n}{1+x} mathrm{d}x = 0$$
$endgroup$
Note that for all $n$ we have :
$$forall xin [0,1], frac{x^n}{1+x} leq 1$$
A constant function is clearly integrable, hence we can apply the dominated convergence theorem to get :
$$lim_{n to infty}int_0^1 frac{x^n}{1+x} mathrm{d}x = int_0^1, lim_{n to infty} frac{x^n}{1+x} mathrm{d}x = 0$$
answered Jan 24 at 21:49
ThinkingThinking
1,13916
1,13916
add a comment |
add a comment |
$begingroup$
Proceeding as in the OP, note that we have two cases
$$frac{1+x^n}{1+x}=begin{cases}sum_{k=1}^{n} (-1)^{k-1}x^{k-1}&,n,,text{odd}\\
frac2{1+x}-sum_{k=1}^{n}(-1)^{k-1}x^{k-1}&,n,,text{even}end{cases}$$
Integrating over $[0,1]$, letting $nto infty$, and using the Taylor series representation for $log(1+x)$ for $x=1$ reveals
$$lim_{ntoinfty}int_0^1 frac{1+x^n}{1+x},dx=log(2) $$
Hence, we have
$$lim_{ntoinfty}int_0^1 frac{x^n}{1+x},dx=log(2)-log(2)=0$$
Rather than proceed as in the OP, we write $frac1{1+x}=sum_{m=0}^{infty}(-x)^m$. Then, we see that
$$begin{align}
int_0^1 frac{x^n}{1+x} ,dx&=sum_{m=0}^{infty} (-1)^m int_0^1 x^{n+m},dx\\
&=sum_{m=n+1}^{infty}(-1)^{m-n-1} frac1{m}tag1
end{align}$$
As the series $sum_{m=1}^{infty}frac{(-1)^{m-n-1}}{m}$ converges, the series on the right hand side of $(1)$ approaches $0$ as $nto infty$.
$endgroup$
$begingroup$
Please let me know how I can improve my answer. I really want to give you the best answer I can. And Happy New Year! ;-)
$endgroup$
– Mark Viola
Jan 30 at 5:04
add a comment |
$begingroup$
Proceeding as in the OP, note that we have two cases
$$frac{1+x^n}{1+x}=begin{cases}sum_{k=1}^{n} (-1)^{k-1}x^{k-1}&,n,,text{odd}\\
frac2{1+x}-sum_{k=1}^{n}(-1)^{k-1}x^{k-1}&,n,,text{even}end{cases}$$
Integrating over $[0,1]$, letting $nto infty$, and using the Taylor series representation for $log(1+x)$ for $x=1$ reveals
$$lim_{ntoinfty}int_0^1 frac{1+x^n}{1+x},dx=log(2) $$
Hence, we have
$$lim_{ntoinfty}int_0^1 frac{x^n}{1+x},dx=log(2)-log(2)=0$$
Rather than proceed as in the OP, we write $frac1{1+x}=sum_{m=0}^{infty}(-x)^m$. Then, we see that
$$begin{align}
int_0^1 frac{x^n}{1+x} ,dx&=sum_{m=0}^{infty} (-1)^m int_0^1 x^{n+m},dx\\
&=sum_{m=n+1}^{infty}(-1)^{m-n-1} frac1{m}tag1
end{align}$$
As the series $sum_{m=1}^{infty}frac{(-1)^{m-n-1}}{m}$ converges, the series on the right hand side of $(1)$ approaches $0$ as $nto infty$.
$endgroup$
$begingroup$
Please let me know how I can improve my answer. I really want to give you the best answer I can. And Happy New Year! ;-)
$endgroup$
– Mark Viola
Jan 30 at 5:04
add a comment |
$begingroup$
Proceeding as in the OP, note that we have two cases
$$frac{1+x^n}{1+x}=begin{cases}sum_{k=1}^{n} (-1)^{k-1}x^{k-1}&,n,,text{odd}\\
frac2{1+x}-sum_{k=1}^{n}(-1)^{k-1}x^{k-1}&,n,,text{even}end{cases}$$
Integrating over $[0,1]$, letting $nto infty$, and using the Taylor series representation for $log(1+x)$ for $x=1$ reveals
$$lim_{ntoinfty}int_0^1 frac{1+x^n}{1+x},dx=log(2) $$
Hence, we have
$$lim_{ntoinfty}int_0^1 frac{x^n}{1+x},dx=log(2)-log(2)=0$$
Rather than proceed as in the OP, we write $frac1{1+x}=sum_{m=0}^{infty}(-x)^m$. Then, we see that
$$begin{align}
int_0^1 frac{x^n}{1+x} ,dx&=sum_{m=0}^{infty} (-1)^m int_0^1 x^{n+m},dx\\
&=sum_{m=n+1}^{infty}(-1)^{m-n-1} frac1{m}tag1
end{align}$$
As the series $sum_{m=1}^{infty}frac{(-1)^{m-n-1}}{m}$ converges, the series on the right hand side of $(1)$ approaches $0$ as $nto infty$.
$endgroup$
Proceeding as in the OP, note that we have two cases
$$frac{1+x^n}{1+x}=begin{cases}sum_{k=1}^{n} (-1)^{k-1}x^{k-1}&,n,,text{odd}\\
frac2{1+x}-sum_{k=1}^{n}(-1)^{k-1}x^{k-1}&,n,,text{even}end{cases}$$
Integrating over $[0,1]$, letting $nto infty$, and using the Taylor series representation for $log(1+x)$ for $x=1$ reveals
$$lim_{ntoinfty}int_0^1 frac{1+x^n}{1+x},dx=log(2) $$
Hence, we have
$$lim_{ntoinfty}int_0^1 frac{x^n}{1+x},dx=log(2)-log(2)=0$$
Rather than proceed as in the OP, we write $frac1{1+x}=sum_{m=0}^{infty}(-x)^m$. Then, we see that
$$begin{align}
int_0^1 frac{x^n}{1+x} ,dx&=sum_{m=0}^{infty} (-1)^m int_0^1 x^{n+m},dx\\
&=sum_{m=n+1}^{infty}(-1)^{m-n-1} frac1{m}tag1
end{align}$$
As the series $sum_{m=1}^{infty}frac{(-1)^{m-n-1}}{m}$ converges, the series on the right hand side of $(1)$ approaches $0$ as $nto infty$.
edited Jan 25 at 1:01
answered Jan 24 at 22:03
Mark ViolaMark Viola
133k1277176
133k1277176
$begingroup$
Please let me know how I can improve my answer. I really want to give you the best answer I can. And Happy New Year! ;-)
$endgroup$
– Mark Viola
Jan 30 at 5:04
add a comment |
$begingroup$
Please let me know how I can improve my answer. I really want to give you the best answer I can. And Happy New Year! ;-)
$endgroup$
– Mark Viola
Jan 30 at 5:04
$begingroup$
Please let me know how I can improve my answer. I really want to give you the best answer I can. And Happy New Year! ;-)
$endgroup$
– Mark Viola
Jan 30 at 5:04
$begingroup$
Please let me know how I can improve my answer. I really want to give you the best answer I can. And Happy New Year! ;-)
$endgroup$
– Mark Viola
Jan 30 at 5:04
add a comment |
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