Let $n$ be a natural number and let $0\lt x\lt{\pi}$. Then, here are my questions.
Question 1: Is the following true?
$$\sum_{k=1}^{n}\frac{\cos(kx)}{k}\gt -1$$
Question 2: Is the following true?
$$\sum_{k=1}^{n}\frac{\sin(kx)}{k}\gt0$$
TEOREMA (L.Fejer-1910, D. Jackson-1912, T.J.Gronwall- 1912). Pentru $x\in (0,\pi)$ au loc inegalitatile
\[(1)\; \; \; \; \; \; \begin{array}{|c|}\hline \\ \sum_{k=1}^{n}\frac{\sin{kx}}{k}> 0 \\ \\ \hline \end{array}\; \; ,\; \; \forall n \in{\mathbb N}. \]
DEMONSTRATION I.
By performing some elementary calculations we find
\ [\ frac {d} {d \ varphi} \ left \ {\ frac {\ sin {2k \ phi}} {(\ sin {\ phi}) ^ {2k}} \ right \} = - 2k \ frac {\ sin {(2k-1) \ phi}} {(\ sin {\ phi}) ^ {2k + 1}} \; . \]
Performing calculations as well as substituting $ \ phi \ leadsto \ frac {x} {2} \; , \; $ by integration we have
\ [\ frac {\ sin {kx}} {k} = 2 \ left (\ sin {\ frac {x} {2}} \ right) ^ {2k} \ int _ {\ frac {x} {2}} ^ {\ frac {\ pi} {2}} \ frac {\ sin {(2k-1) \ phi}} {(\ sin \ phi) ^ {2k + 1}} \; d \ phi \; \; , \; k \ in {\ mathbb N}. \]
Sum for $ k \ in \ {1,2, ..., n \} $ is obtained
\ [\ sum_ {k = 1} ^ {n} \ frac {\ sin {kx}} {k} = 2 \ int _ {\ frac {x} {2}} ^ {\ frac {\ pi} {2}} \ sum_ {k = 1} ^ {n} \ left [r (x, \ phi) \ right] ^ {k} \ frac {\ sin {(2k-1) \ phi}} {\ sin {\ phi}} \; d \ phi \]
unde \ [r (x, \ phi): = \ left (\ frac {\ sin \ frac {x} {2}} {\ sin {\ phi}} \ right) ^ {2} \ in [0,1] \]
for $ 0 \ le \ frac {x} {2} \ le \ phi <\ frac {\ pi} {2} \; . $ The above equation implies
\ [\ sum_ {k = 1} ^ {n} \ frac {\ sin {kx}} {k} = \ int_ {x} ^ {\ pi} \ sum_ {k = 1} ^ {n} \ left [r (x, \ frac {\ psi} {2}) \ right] ^ {k} \ frac {\ sin {\ left (k- \ frac {1} {2} \ right) \ psi}} {\ sin {\ frac {\ psi} {2}}} \; d \ psi \; . \]
Applying Abel's identity ("'summation by parts"), i.e.
\ [\ sum_ {k = 1} ^ {n} A_ {k} B_ {k} = \ sum_ {k = 1} ^ {n-1 } \ left (A_ {k} -A_ {k + 1} \ right) \ sum_ {j = 1} ^ {k} B_ {j} + A_ {n} \ sum_ {j = 1} ^ {n} B_ {j} \; , \]
and noting
\[\left\{\begin{array}{rcl}{\mathbf r}&: =&r(x,\frac{\psi}{2})\; \; , \; \;{\mathbf r}\in [0,1]\; \; \; \mbox{vezi}\; \; ()\\{\mathcal F}_{n}(\psi)&: =&\sum_{k=1}^{n}\sin{\left(k-\frac{1}{2}\right)\psi}\; \; \; , \; \;{\mathcal F}_{n}(\psi)\ge 0 \; \; \mbox{dac'a}\; x\in [0,\pi]\; \;-\mbox{vezi L. Fej\'er}\end{array}\right.\; , \] din identitatea lui Abel g'asim
\[\sum_{k=1}^{n}\frac{\sin{kx}}{k}=\int_{x}^{\pi}\left\{(1-{\mathbf r})\sum_{k=1}^{n-1}{\mathbf r}^{k-1}{\mathcal F}_{k}(\psi)+{\mathbf r}^{n}{\mathcal F}_{n}(\psi)\right\}\frac{d\psi}{\sin{\frac{\psi}{2}}}\; . \]
Deoarece ${\mathcal F}_{n}(\psi)=\frac{1-\cos(n\psi)}{2\sin{\frac{\psi}{2}}}\; ,$ concludem cu
\[\begin{array}{|c|}\hline \\ \sum_{k=1}^{n}\frac{\sin{kx}}{k}=\frac{1}{2}\int_{x}^{\pi}\left\{(1-{\mathbf r})\sum_{k=1}^{n-1}{\mathbf r}^{k-1}\left(1-\cos(k\psi)\right)+{\mathbf r}^{n}(1-\cos(n\psi))\right\}\frac{d\psi}{\left(\sin{\frac{\psi}{2}}\right)^{2}}\\ \\ \hline \end{array}\; . \]
Aceasta reprezentare completeaza demonstra'tia I.
DEMONSTRATIA II.
Fie $P_{n}(x)$ polinomul lui Legendre de gradul $n$, adic'a
\[\begin{array}{c}P_{n}(x)=\frac{1}{n! 2^{n}}\left[(x^{2}-1)^{n}\right)^{(n)}={}_{2}F_{1}\left(-n,n+1;1;\frac{1-x}{2}\right)=\\ \\ =\frac{1}{2^{n}}\sum_{k=0}^{\lfloor \frac{n}{2}\rfloor}{n\choose 2k}(-1)^{k}\frac{(n+1)_{n-2k}(2k)!}{k!(n-k)!}x^{n-2k}\end{array}\]
unde
\[\begin{array}{c}{}_{2}F_{1}(-n;b;c;z): =\sum_{k=0}^{n}(-1)^{k}{n\choose k}\frac{(b)_{k}}{(c)_{k}}z^{k}\\ (d)_{k}: =d(d+1)\cdots (d+k-1)\; ,\; \; k\in{\mathbb N}\; ,\; (d)_{0}: =1 . \end{array}\]
Se cunosc urmatoarele:
-- radacinile lui $P_{n}(x)$ sunt reale,distincte, situate in $(-1,1)$;
-- $|P_{n}(t)|\le 1 \; ,\; \; \forall t\in [-1,1] .$
Demonstration II (see [16] as well as R. Askey's comments)
is based on the identity:
\ [\ begin {array} {| c |} \ hline \\ \ sum_ {k = 1} ^ {n} \ frac {\ sin (k \ cdot \ arccos {x})} {k} = \ frac {\ sqrt {1-x}} {2} \ int _ {- 1} ^ {x} \ frac {1-P_ {n} (y)} {1-y} \ frac {dy} {\ sqrt {xy}} \\ \\ \ hline \ end {array} \; , \; x \ in (-1,1). \; . \]
Remarks.
1) Inequality (1) was conjectured by Leopold (Lipot) Fejer. It was later demonstrated by D. Jackson- [14] and THGronwall- [12].
There are currently over 100 demonstrations. We have chosen to present to you those which the undersigned considers "simpler".
It is said that until his death L.Fejer sought to find new demonstrations of his (1). L. Fejer also taught at the University of Cluj (under the name of L. Weiss).
2) Although simple at first glance, inequality (1) has given the mathematicians a fork.
3) Inequality (1) occurs in the following areas: Fourier series (Gibbs phenomenon - see [13]), Orthogonal polynomials ([1] - [5], [18]), Complex functions (Demonstration of Bieberbach's conjecture, univalent functions , [4], [6]), Theory of Approximation ([5]).
BIBLIOGRAFIE.
[1]R. Askey , Orthogonal Polynomials and Special Functions,
Regional Conf.Lect.Appl.Math., vol.21,SIAM,Philadelphia,Pa., 1975.
[2]R.Askey Positive quadrature methods and positive polynomial sums, 'in Approximation Theory V, Academic Press, 1986.
[3] R. Askey and J. Fitch , Integral reprezentations for Jacobi
polynomial amd some applications ,J.Math.Anal.Appl., 26 (1969)
411-437.
[4]R. Askey and G. Gasper , Positive Jacobi polynomial sums,(II),
Amer.J.Math., 98 (1976) 709-737.
[4]R. Askey and G. Gasper , Inequalities for polynomials}, 'in "The Bieberbach Conjecture", Proc.of the Symposium on the Occasion
of the Proof, Mathematical Surveys and Monographs, 21, Amer.Mathematical Society, 1986, 7-32.
[5]H. Bavinck , Jacobi Series and Approximation,
Mathematical Centre Tracts 39, Mathematisch Centrum Amsterdam 1972.
[6] L. de Branges , The Story of the Verification of the Bieberbach Conjecture, 'in " The Bieberbach Conjecture", Proc.of the Symposium on the Occasion of the Proof, Mathematical Surveys and Monographs, 21, Amer.Mathematical Society, 1986, 199-203.
[7] L. Fejer , Sur les fonctions bornees et integrables,
C.R.Acad.Sci.Paris 131 (1900) 984-987.
[8] L. Fejer , Sur le develpopment d'une fonction arbitraire suivant les fonctions de Laplace, C.R.Acad.Sci.Paris , 146 (1908) 224-227.
[9]L. Fejer , Ueber die Laplacesche Reihe, Math. Ann. (1909)76-109.
[10] L. Fejer, Some theorems relating to the sign of an entire rational function, etc., Monta.fuer Math. und Phys., 35 (1928) 305-344.
[11]L. Fejer, Collected Works (I)-(II), Birkhauser Verlag, Basel, 1970.
[12]TH Gronwall , On the Gibbs phenomenon and the trigonometric sums $\sin{x}+(1/2)\sin{2x}+...+(1/n)\sin{nx}$ , Math. Ann., 72 (1912) 228-243.
[13] E. Hewitt and RE Hewitt, The Gibbs-Wilbraham phenomenon: an epsiode in Fourier analysis, Arch.Hist.Exact Sci., 21 (1979) 129-160.
[14]D. Jackson, On a Trigonometric Sum, Rend.Circ.Mat.Palermo 32(1911) 257-262.
[15]A. Lupas, Advanced Problem 6517, Amer.Math.Monthly (1986) p. 305 ; (1988) p.264.
[16]A. Lupas, Advanced Problem 6585, Amer.Math.Monthly
(1988) p. 880; (1990) p.859-860.
[17]L. Lupas , An identity for ultraspherical polynomials, Revue d${}^{,}$Numerical analysis and Theorie de l'approximation, tome 24 , no.1-2 (1995) 181-185.
[18]G. Szego, Orthogonal Polynomials, Amer.Math.Soc.Colloq. Publications vol.23, fourth ed., Amer.Math.Soc., Providence, RI, 1975.
Iron-up inequality $ \ sum_ {k = 1} ^ {n} \ frac {\ sin {kx}} {k}> x \ left (1- \ frac {x} {\ pi} \ right) ^ 3 $ holds for $ x \ in (0, \ pi). $
**In short:** let $f_n(x)$ denote the function on the lhs of the inequality. Of course, $f_1(x)=\sin x\geq 0$ on $[0,\pi]$. We will prove that $f_n(x)\geq 0$ on $[0,\pi]$ by induction on $n$. It is not too hard to determine the local minima of $f_n$ on $[0,\pi]$ by investigating its derivative. Then Ma Ming observed that $f_n$ coincides with $f_{n-1}$ on these local minima. And the induction step follows easily. Of course, $f_n(0)=f_n(\pi)=0$. We will actually prove that
>$$
f_n(x)=\sum_{k=1}^n\frac{\sin kx}{k}>0\qquad\forall x\in(0,\pi).
$$
**Remark:** it is worth noting that the $f_n$'s are the partial sums of the Fourier series of the same sawtooth function. Just [look at the case $n=6$][1], for instance, to see how they tend to approximate it nicely. [See here][2] to get an idea how to estimate the error in such approximations. As pointed out by math110, there are many proofs of this so-called Fejer-Jackson inequality. It can even be shown that the [$f_n$'s are bounded below][3] by a certain nonnegative polynomial on $[0,\pi]$. The proof below is at the calculus I level. I'm not sure it can be made more elementary.
**Proof:** first, $f_1(x)=\sin x$ is positive on $(0,\pi)$. Assume this holds for $f_{n-1}$ for some $n\geq 2$. Then observe that $f_n$ is differenbtiable on $\mathbb{R}$ with
$$
f_n'(x)=\sum_{k=1}^n\cos kx=\mbox{Re} \sum_{k=1}^n (e^{ix})^k.
$$
For $x\in 2\pi \mathbb{Z}$, we have $f_n'(x)=n$. So the zeros of $f_n'$ are the zeros of
$$
\mbox{Re}\;e^{ix}\frac{e^{inx}-1}{e^{ix}-1}=\mbox{Re}\;e^{i(n+1)x/2}\frac{\sin (nx/2)}{\sin(x/2)}=\frac{\cos((n+1)x/2)\sin (nx/2)}{\sin(x/2)}.
$$
This yields
$$
\frac{nx}{2}\in \pi\mathbb{Z}\quad\mbox{or}\quad \frac{(n+1)x}{2}\in \frac{\pi}{2}+\pi\mathbb{Z}
$$
i.e.
$$
x\in \frac{2\pi}{n}\mathbb{Z}\quad\mbox{or}\quad x\in \frac{\pi}{n+1}+\frac{2\pi}{n+1}\mathbb{Z}.
$$
Between $0$ and $\pi$, these are ordered as follows:
$$
0<\frac{\pi}{n+1}<\frac{2\pi}{n}<\frac{3\pi}{n+1}<\frac{4\pi}{n}<\ldots < \frac{2\lfloor n/2\rfloor \pi}{n}\leq \pi.
$$
The sign of $f_n'$ changes at each of these zeros, starting from a positive sign on $(0,\pi/(n+1))$. It follows that $f_n$ is positive on the latter, positive on the last interval (if nontrivial, i.e. in the odd case), with local minima at
$$\frac{2j\pi}{n}\qquad\mbox{for}\qquad j=1,\ldots,\lfloor n/2\rfloor.$$
But now here is Ma Ming's key observation: for these values, we have
$$
f_n\left(\frac{2j\pi}{n}\right)=f_{n-1}\left(\frac{2j\pi}{n}\right)+\sin\left(n\cdot\frac{2j\pi}{n}\right)=f_{n-1}\left(\frac{2j\pi}{n}\right)>0
$$
by induction step. It follows that $f_n(x)>0$ on $(0,\pi)$. QED.
[1]: http://www.wolframalpha.com/input/?i=sin%20%28x%29%2bsin%20%282x%29/2%2bsin%20%283x%29/3%2bsin%20%284x%29/4%2bsin%285x%29/5%2bsin%286x%29/6
[2]: https://math.stackexchange.com/questions/57054/asymptotic-error-of-fourier-series-partial-sum-of-sawtooth-function
[3]: https://math.stackexchange.com/questions/177995/a-pseudo-fejer-jackson-inequality-problem
References: Forum 1 , Forum 1.1 and here