I have this problem which says that for any positive integer $n$, $n \neq 0$ the following inequality is true: $$\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{3^3}\right)\cdots\left(1+\frac{1}{n^3}\right) < 3$$ This problem was given to me in a lecture about induction but any kind of solution would be nice.And also I'm in 10th grade :)
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5Can you show us your attempt? You will be more likely to get a response. – Sam Streeter Oct 25 '18 at 15:03
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2Maple gives us this term for your product: $${\frac {\cosh \left( 1/2,\pi,\sqrt {3} \right) \left( n+1 \right) \Gamma \left( n+1/2+i/2\sqrt {3} \right) \Gamma \left( n+1/2-i/2\sqrt {3} \right) }{\pi, \left( \Gamma \left( n+1 \right) \right) ^{2}}} $$ – Dr. Sonnhard Graubner Oct 25 '18 at 15:09
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2I suspect someone will prove the $n\to\infty$ limit is $\pi^{-1}\cosh\frac{\pi\sqrt{3}}{2}$. – J.G. Oct 25 '18 at 15:19
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@Dr.SonnhardGraubner Thanks for your comment but I forgot to mention that I'm in highschool, 10th grade so have no idea about that. – LIR Oct 25 '18 at 15:21
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1@J.G. You were right. – Oct 25 '18 at 16:36
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1Does the OP have a beef with somebody? It seems that somebody (most likely a single person) downvoted the OP and almost everybody who answered here at about the same time (i.e., about 9 hours before my comment here), even though all answers are sensible. – Batominovski Oct 26 '18 at 10:28
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2Possible duplicate of Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$ – Sil Nov 05 '18 at 08:04
8 Answers
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The cases $n=1$ and $n=2$ can be verified manually. We assume that $n\geq 3$. For an integer $k>2$, we have $$1+\frac{1}{k^3}=\left(1+\frac1k\right)\left(1-\frac1k+\frac1{k^2}\right)=\left(1+\frac1k\right)\left(\frac{k-1}{k}\right)^2\left(1+\frac{1}{k-1}+\frac{1}{(k-1)^2}\right).$$ We note that $$1+\frac{1}{k-1}+\frac1{(k-1)^2}=\frac{1-\frac{1}{(k-1)^3}}{1-\frac{1}{k-1}}<\frac{1}{1-\frac{1}{k-1}}=\frac{k-1}{k-2}$$ for $k>2$. That is, $$\prod_{k=1}^n\left(1+\frac1{k^3}\right)\leq \left(1+\frac1{1^3}\right)\left(1+\frac1{2^3}\right)\prod_{k=3}^n\left(1+\frac1k\right)\left(\frac{k-1}{k}\right)^2\left(\frac{k-1}{k-2}\right).$$ The RHS can be telescoped nicely: $$\prod_{k=3}^n\left(1+\frac1k\right)=\frac{n+1}{3},$$ $$\prod_{k=3}^n\left(\frac{k-1}{k}\right)=\frac{2}{n},$$ and $$\prod_{k=3}^n\left(\frac{k-1}{k-2}\right)=n-1.$$ Thus, $$\prod_{k=1}^n\left(1+\frac1{k^3}\right)\leq 2\left(\frac98\right)\left(\frac{n+1}{3}\right)\left(\frac{2}{n}\right)^2(n-1)=3\left(\frac{n^2-1}{n^2}\right)<3.$$
In fact, for a fixed positive integer $m\geq 3$ and for every $n\geq m$, we have $$\prod_{k=1}^n\left(1+\frac1{k^3}\right)\leq t_m\left(\frac{n^2-1}{n^2}\right) <t_m,$$ where $$t_m=\frac{m^2}{m^2-1}\ \prod_{k=1}^m\left(1+\frac{1}{k^3}\right).$$ If we pick $m=5$, we get $m=\frac{637}{256}<\frac{640}{256}=\frac52$. So, we can prove a stronger inequality $$\prod_{k=1}^n\left(1+\frac1{k^3}\right)<\frac52.$$
By writing $1+\frac{1}{k^3}$ as $$\frac{(k+1)(k-u)(k-v)}{k^3}=\frac{k+1}{k}\left(\frac{\Gamma(k+1-u)}{\Gamma(k-u)}\right)\left(\frac{\Gamma(k+1-v)}{\Gamma(k-v)}\right)\left(\frac{\Gamma(k)}{\Gamma(k+1)}\right)^2,$$ where $u=\frac{1+\sqrt{3}i}2$ and $v=\bar{u}=1-u$, we have $$\prod_{k=1}^n\left(1+\frac1{k^3}\right)=\frac{(n+1)\Gamma(n+1-u)\Gamma(n+1-v)}{\Gamma(1-u)\Gamma(1-v)\big(\Gamma(n+1)\big)^2}.$$ From the reflection formula, and from the relation $u+v=1$, we have $$\Gamma(1-u)\Gamma(1-v)=\Gamma(1-u)\Gamma(u)=\left(\frac{\pi}{\sin(\pi u)}\right).$$ Since \begin{align}\sin(\pi u)&=\sin\left(\frac{\pi}{2}+\frac{\sqrt{3}}{2}\pi i\right)\\&=\sin\left(\frac{\pi}{2}\right)\cosh\left(\frac{\sqrt{3}}{2}\pi\right)+i\cos\left(\frac{\pi}{2}\right)\sinh\left(\frac{\sqrt{3}}{2}\pi\right) \\&=\cosh\left(\frac{\sqrt{3}}{2}\pi\right),\end{align} we conclude that $$\Gamma(1-u)\Gamma(1-v)=\frac{\pi}{\cosh\left(\frac{\sqrt{3}}{2}\pi\right)}.$$ This leads to \begin{align}\prod_{k=1}^n\left(1+\frac1{k^3}\right)&=\frac{\cosh\left(\frac{\sqrt{3}}{2}\pi\right)}{\pi}\left(\frac{(n+1)\Gamma(n+1-u)\Gamma(n+1-v)}{\big(\Gamma(n+1)\big)^2}\right) \\&=\frac{\cosh\left(\frac{\sqrt{3}}{2}\pi\right)}{\pi}\left(\frac{\Gamma(n+1-u)\ (n+1)^u}{\Gamma(n+1)}\right)\left(\frac{\Gamma(n+1-v)\ (n+1)^v}{\Gamma(n+1)}\right).\end{align} Since $$\lim_{n\to\infty}\frac{\Gamma(n+z)}{\Gamma(n)\ n^z}=1$$ for all $z\in\mathbb{C}$, we get $$\prod_{k=1}^\infty\left(1+\frac1{k^3}\right)=\frac{\cosh\left(\frac{\sqrt{3}}{2}\pi\right)}{\pi}\approx 2.42818979.$$
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But if you can tell me what was the thought process for factoring $1+\frac{1}{k^3}$ that way? – LIR Oct 25 '18 at 15:51
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5First I factorized $1+\frac1{k^3}=\left(1+\frac1k\right)\left(1-\frac1k+\frac1{k^2}\right)$. I wanted to somehow rewrite $1-\frac1k+\frac1{k^2}$ as something of the form $1+a+a^2$ for $0<a<1$, so that I can use $$1+a+a^2<1+a+a^2+a^3+\ldots=\frac{1}{1-a}.$$ So I observed that $$1-\frac1k+\frac1{k^2}=\frac{k^2-k+1}{k^2}=\frac{(k-1)^2+(k-1)+1}{k^2}.$$ If I changed the denominator to be $(k-1)^2$, then I would get $$\frac{(k-1)^2+(k-1)+1}{(k-1)^2}=1+\frac{1}{k-1}+\frac{1}{(k-1)^2},$$ which is what I need. That is why you have the "rescaling term" $\left(\frac{k-1}{k}\right)^2$. – Oct 25 '18 at 15:54
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4I wish I could upvote once for the inequality and again for the exact calculation. – saulspatz Oct 25 '18 at 16:42
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Denote $p(n):=(1+\frac{1}{1^3})(1+\frac{1}{2^3})(1+\frac{1}{3^3})...(1+\frac{1}{n^3})$.
Claim: $$p(n)\leq3-\frac2{n^2},\,\forall n\geq2.$$
For $n=2$, we have $\frac94\leq3-\frac12$.
Then suppose $p(n)\leq3-\frac2{n^2}$ for some $n$. We see $$\eqalign{ p(n+1)&=p(n)(1+\frac1{(n+1)^3})\cr &\leq3-\frac{2}{n^2}+\frac3{(n+1)^3}-\frac{2}{n^2(n+1)^3}\cr &=3+\frac{3n^2-2(n^3+3n^2+3n+1)-2}{n^2(n+1)^3}\cr &=3-\frac{2n^3+3n^2+6n+4}{n^2(n+1)^3}\cr &=3-\frac{2n^3+2n^2+(n^2+6n+4)}{n^2(n+1)^3}\cr &\leq3-\frac{2n^2(n+1)}{n^2(n+1)^3}\cr &\leq3-\frac2{(n+1)^2}}.$$
As pointed out by @saulspatz, one can prove that $p(n)\leq3-\frac1n,\forall n\geq1$ by the same method.
Hope this helps.
awllower
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+1 If I'm not mistaken, you can prove $p(n)\geq3-{1\over n}, n\geq1$ by the same method an a little simpler algebra. – saulspatz Oct 25 '18 at 17:11
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@saulspatz I think you mean $p(n)\leq3-\frac1n$. Indeed this is a sharper result and easier to prove. Thanks for pointing this out! – awllower Oct 26 '18 at 00:06
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@awllower Yes, that was a typo, but it's too late to edit the comment. – saulspatz Oct 26 '18 at 04:35
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We have that
$$\prod_{k=1}^\infty \left(1+\frac{1}{k^3}\right)<3\iff \sum_{k=1}^\infty \log\left(1+\frac{1}{k^3}\right)<\log 3$$
and since $\forall x>0\, \log(1+x)<x$
$$\sum_{k=1}^\infty \log\left(1+\frac{1}{k^3}\right)=\log 2+\sum_{k=2}^\infty \log\left(1+\frac{1}{k^3}\right)<\log 2+\sum_{k=2}^\infty \frac{1}{k^3}<\log 3$$
user
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Taking seriously the suggestions of Dr. Sonnhard Graubner and J.G., one can indeed prove that, for $x\in\mathbb{C}$ and $p\in\mathbb{N}_+$, $$ \prod_{n=1}^\infty \left(1+\frac{x^p}{n^p}\right) = \prod_{j=1}^p \frac{1}{\Gamma(1+\omega_p^j x)} , $$ where $\{-\omega_p^j\}_{j=1}^p$ are the $p$-th roots of $-1$.
In fact, using that $\sum_{j=1}^p \omega_p^j = 0$ and this identity (see also this), one has that $$ \prod_{n=1}^N \left(1+\frac{x^p}{n^p}\right) = \prod_{j=1}^p \prod_{n=1}^N \left(1+\frac{\omega_p^j x}{n}\right) = \prod_{j=1}^p e^{-\omega_p^j x \gamma} \prod_{n=1}^N \left(1+\frac{\omega_p^j x}{n}\right) e^{-\frac{\omega_p^j x}{n}} $$ converges, as $N\to\infty$, to $$ \prod_{j=1}^p \frac{1}{\omega_p^j x \Gamma(\omega_p^j x)} = \prod_{j=1}^p \frac{1}{\Gamma(1+\omega_p^j x)}. $$
Specializing for $p=3$ and $x=1$ and using the formula for the absolute value here, we have $$ \prod_{n=1}^\infty \left(1+\frac{1}{n^p}\right) = \frac1{\Gamma(1+1)\Gamma(1-\frac12-\frac{\sqrt3 i}2)\Gamma(1-\frac12+\frac{\sqrt3 i}2)} = \frac1{|\Gamma(\frac12+\frac{\sqrt3 i}2)|} = \frac{\cosh\left(\frac{\sqrt3 }2\pi\right)}{\pi} . $$
Federico
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2Very nice. Indeed, if $x_1,x_2,\ldots,x_m$ and $y_1,y_2,\ldots,y_m$ are complex numbers which are not positive integers such that $x_1+x_2+\ldots+x_m=y_1+y_2+\ldots+y_m$. Then, $$\prod_{n=1}^\infty,\prod_{j=1}^m,\left(\frac{n-x_j}{n-y_j}\right)=\prod_{j=1}^m,\left(\frac{\Gamma\left(1-y_j\right)}{\Gamma\left(1-x_j\right)}\right),.$$ – Batominovski Oct 26 '18 at 10:47
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2On the other hand, if $\displaystyle\text{Re}\left(\sum_{j=1}^m,x_j-\sum_{j=1}^m,y_j\right)>0$, then $\displaystyle\prod_{n=1}^\infty,\prod_{j=1}^m,\left(\frac{n-x_j}{n-y_j}\right)=0$, and if $\displaystyle\text{Re}\left(\sum_{j=1}^m,x_j-\sum_{j=1}^m,y_j\right)<0$, then $\displaystyle\prod_{n=1}^\infty,\prod_{j=1}^m,\left(\frac{n-x_j}{n-y_j}\right)=\infty$. – Batominovski Oct 26 '18 at 10:51
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2If $\displaystyle\text{Re}\left(\sum_{j=1}^m,x_j-\sum_{j=1}^m,y_j\right)=0$, but $\displaystyle\sum_{j=1}^m,x_j-\sum_{j=1}^m,y_j\neq 0$, then $\displaystyle\prod_{n=1}^\infty,\prod_{j=1}^m,\left(\frac{n-x_j}{n-y_j}\right)$ does not converge at all in $\mathbb{C}\cup{\infty}$. – Batominovski Oct 26 '18 at 10:56
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With $AM-GM$ \begin{align} 1.(1+\frac{1}{2^3})(1+\frac{1}{3^3})\cdots(1+\frac{1}{n^3}) &\leq\left(\dfrac1n(n+\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+\cdots+\frac{1}{n^3})\right)^n \\ &\leq\left(\dfrac1n(n+\sum_{n=1}^\infty\frac{1}{n^3}-1)\right)^n \\ &\leq\left(\dfrac1n(n+\zeta(3)-1)\right)^n \\ &\leq\left(1+\dfrac{\zeta(3)-1}{n}\right)^n \\ &< e^{\zeta(3)-1}\\ &<\frac32 \end{align} Thanks to Winther.
Nosrati
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By induction we can prove the stronger
$$\prod_{k=1}^n\left(1+\frac{1}{k^3}\right)<3\left(1-\frac{1}{n}\right)$$
indeed
1. base cases: by inspection the inequality is satisfied for for $n=1,2, 3$
2. induction step:
assume true that (Ind. Hyp.): $\prod_{k=1}^n\left(1+\frac{1}{k^3}\right)<3\left(1-\frac{1}{n}\right)$
we want to prove that: $\prod_{k=1}^{n+1}\left(1+\frac{1}{k^3}\right)<3\left(1-\frac{1}{n+1}\right)$
then we have
$$\prod_{k=1}^{n+1}\left(1+\frac{1}{k^3}\right)=\prod_{k=1}^n\left(1+\frac{1}{k^3}\right) \cdot \left(1+\frac{1}{(n+1)^3}\right)<$$ $$\stackrel{Ind. Hyp.}<3\left(1-\frac{1}{n}\right)\left(1+\frac{1}{(n+1)^3}\right)\stackrel{?}<3\left(1-\frac{1}{n+1}\right)$$
thus we need to show that
$$3\left(1-\frac{1}{n}\right)\left(1+\frac{1}{(n+1)^3}\right)\stackrel{?}<3\left(1-\frac{1}{n+1}\right)$$
which is true indeed
$$1+\frac{1}{{n+1}^3}-\frac1n-\frac{1}{n(n+1)^3}\stackrel{?}<1-\frac{1}{n+1}$$
$$n-(n+1)^3-1\stackrel{?}<-n(n+1)^2$$
$$n-n^3-3n^2-3n-1-1\stackrel{?}<-n^3-2n^2-n$$
$$n^2+n+2\stackrel{?}>0$$
user
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@LazarIonutRadu I just want to give an answer to your original question for a method by induction, it is interesting since how you can see we need to consider a stronger inequality to can proceed by induction. It is a typical case we can encounter with the method. – user Oct 25 '18 at 16:36
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Claim:
For all positive integers $n$, we have $$\prod_{k=1}^n \left(1+\frac{1}{k^3}\right) < e$$ or equivalently $$\sum_{k=1}^n \ln\left(1+\frac{1}{k^3}\right) < 1$$ Proof:
It suffices to show that $$\sum_{k=1}^n \ln\left(1+\frac{1}{k^3}\right) < 1-\frac{1}{(n+1)^2}\tag{*}$$ holds for all positive integers $n$.
To prove $(*)$, proceed by induction on $n$.
By direct evaluation, $(*)$ holds for the base case $n=1$.
Suppose $(*)$ holds for some positive integer $n$. \begin{align*} \text{Then}\;\;&\sum_{k=1}^{n+1} \ln\left(1+\frac{1}{k^3}\right)\\[4pt] =\;& \left( \sum_{k=1}^n \ln\left(1+\frac{1}{k^3}\right) \right) + \ln\left(1+\frac{1}{(n+1)^3}\right) \\[4pt] <\;& \left( 1-\frac{1}{(n+1)^2}\right) + \ln\left(1+\frac{1}{(n+1)^3}\right) &&\text{[by the inductive hypothesis]} \\[4pt] <\;& \left( 1-\frac{1}{(n+1)^2}\right) + \frac{1}{(n+1)^3} &&\text{[since $\ln(1+x) < x$, for all $x > 0$]} \\[4pt] =\;&\left(1-\frac{1}{(n+2)^2}\right)-\frac{n^2+n-1}{(n+1)^3(n+2)^2} \\[4pt] <\;&1-\frac{1}{(n+2)^2} &&\text{[since $n^2+n-1 > 0$]} \\[4pt] \end{align*} which completes the induction, and thus proves the claim.
quasi
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Less Than $\boldsymbol{3}$
The inequality $$ 1+\frac1{n^3}\lt\frac{1+\frac1{2(n-1)^2}}{1+\frac1{2n^2}}\tag1 $$ can be verified by cross-multiplying and then multiplying both sides by $2n^5(n-1)^2$; that is, $$ 2n^7-4n^6+3n^5\underbrace{-3n^3+3n^2-2n+1}_\text{$-(3n^2+2)(n-1)-1\lt0$ for $n\ge1$}\lt2n^7-4n^6+3n^5\tag2 $$ Therefore, employing a telescoping product, $$ \begin{align} \prod_{n=1}^\infty\left(1+\frac1{n^3}\right) &\lt2\prod_{n=2}^\infty\frac{1+\frac1{2(n-1)^2}}{1+\frac1{2n^2}}\\ &=2\cdot\frac32\\[9pt] &=3\tag3 \end{align} $$
Actual Value
$$
\begin{align}
\lim_{n\to\infty}\prod_{k=1}^n\frac{k^3+1}{k^3}
&=\lim_{n\to\infty}\frac{\Gamma(n+2)\,\Gamma\!\left(n+\frac12+i\frac{\sqrt3}2\right)\Gamma\!\left(n+\frac12-i\frac{\sqrt3}2\right)}{\Gamma(2)\,\Gamma\!\left(\frac12+i\frac{\sqrt3}2\right)\Gamma\!\left(\frac12-i\frac{\sqrt3}2\right)\Gamma(n+1)^3}\tag4\\
&=\frac1{\Gamma\!\left(\frac12+i\frac{\sqrt3}2\right)\Gamma\!\left(\frac12-i\frac{\sqrt3}2\right)}\\
&\times\lim_{n\to\infty}\frac{\Gamma(n+2)\,\Gamma\!\left(n+\frac12+i\frac{\sqrt3}2\right)\Gamma\!\left(n+\frac12-i\frac{\sqrt3}2\right)}{\Gamma(n+1)^3}\tag5\\
&=\frac{\sin\left(\frac\pi2+i\frac{\pi\sqrt3}2\right)}{\pi}\times1\tag6\\[6pt]
&=\frac{\cosh\left(\frac{\pi\sqrt3}2\right)}{\pi}\tag7
\end{align}
$$
Explanation:
$(4)$: $\prod\limits_{k=1}^n(k+x)=\frac{\Gamma(n+1+x)}{\Gamma(1+x)}$ and $k^3+1=(k+1)\left(k-\frac12+i\frac{\sqrt3}2\right)\left(k-\frac12-i\frac{\sqrt3}2\right)$
$(5)$: pull out the constant factor using $\Gamma(2)=1$
$(6)$: apply Euler's Reflection Formula $\Gamma(x)\,\Gamma(1-x)=\frac\pi{\sin(\pi x)}$
$\phantom{(6)\text{:}}$ and Gautschi's Inequality, which implies $\lim\limits_{n\to\infty}\frac{\Gamma(n+x)}{\Gamma(n)\,n^x}=1$
$(7)$: $\cos(ix)=\cosh(x)$
robjohn
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