Summation of trigonometric series

[84/70%]
Show
LaTeX
Problem 2551 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.1 "Numerical series, tests for convergence of series of constant sign"

Directly prove the convergence of the following series and find their sums:

?
(a)

qsin⁡α+q2sin⁡2α+⋯+qnsin⁡nα+⋯(∣q∣<1)q \sin \alpha +q^{2} \sin 2\alpha +\cdots +q^{n} \sin n\alpha +\cdots \quad (\left|q\right|<1);

(b)

qcos⁡α+q2cos⁡2α+⋯+qncos⁡nα+⋯(∣q∣<1)q \cos \alpha +q^{2} \cos 2\alpha +\cdots +q^{n} \cos n\alpha +\cdots \quad (\left|q\right|<1).

Problem 3018 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.7 "Methods of Summing Series"

Find the sum of the following trigonometric series:

∑n=1∞sin⁡nxn. \sum _{n=1}^{\infty } \frac{\sin nx}{n}.
?
Problem 3019 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.7 "Methods of Summing Series"

Find the sum of the following trigonometric series:

∑n=1∞cos⁡nxn. \sum _{n=1}^{\infty } \frac{\cos nx}{n}.
?
Problem 3020 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.7 "Methods of Summing Series"

Find the sum of the following trigonometric series:

∑n=1∞sin⁡nαsin⁡nxn. \sum _{n=1}^{\infty } \frac{\sin n\alpha \sin nx}{n}.
?
Problem 3021 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.7 "Methods of Summing Series"

Find the sum of the following trigonometric series:

∑n=1∞sin⁡2nαsin⁡nxn(0<α<π2). \sum _{n=1}^{\infty } \frac{\sin ^{2} n\alpha \sin nx}{n} \quad \left(0<\alpha <\frac{\pi }{2}\right).
?
Problem 3022 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.7 "Methods of Summing Series"

Find the sum of the following trigonometric series:

∑n=1∞sin⁡(2n−1)x2n−1. \sum _{n=1}^{\infty } \frac{\sin (2n-1)x}{2n-1}.
?
Problem 3023 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.7 "Methods of Summing Series"

Find the sum of the following trigonometric series:

∑n=2∞(−1)ncos⁡nxn2−1. \sum _{n=2}^{\infty } (-1)^{n} \frac{\cos nx}{n^{2}-1}.
?
Problem 2.22.48 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove that 3x2−6πx+2π212=∑n=1∞cos⁡nxn2,0⩽x⩽π\frac{3 x^{2}-6 \pi x+2 \pi^{2}}{12} = \sum_{n = 1}^{\infty } \frac{\cos n x}{n^{2}}, 0 \leqslant x \leqslant \pi.

Hint. Use the results of problems 40 and 45.

?
Problem 2.22.56 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove that

∑n=1∞cos⁡(2n−1)x(2n−1)4=196π(π−2x)(π2+2πx−2x2),0⩽x⩽π \sum _{n = 1}^{\infty } \frac{\cos (2 n-1) x}{(2 n-1)^{4}} = \frac{1}{96} \pi (\pi -2 x)\left(\pi ^{2}+2 \pi x-2 x^{2}\right), \quad 0 \leqslant x \leqslant \pi
?
Problem 2.22.60 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Expand the function into a Fourier series using the complex form of the Fourier series:

?
(1)

f(x)=1−a21−2acos⁡x+a2,∣a∣<1f(x) = \frac{1-a^{2}}{1-2 a \cos x+a^{2}},\left|a\right| < 1;

(2)

f(x)=1−acos⁡x1−2acos⁡x+a2,∣a∣<1f(x) = \frac{1-a \cos x}{1-2 a \cos x+a^{2}},\left|a\right| < 1.

Problem 2.22.61 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Expand into a Fourier series the unbounded periodic function:

?
(1)

f(x)=ln⁡∣sin⁡(x/2)∣f(x) = \ln \left|\sin (x / 2)\right|

(2)

f(x)=ln⁡∣tg⁡(x/2)∣f(x) = \ln \left|\operatorname {tg}(x / 2)\right|.

Problem 2.22.63 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Using the expansion of the function ln⁡∣−2cos⁡(x/2)∣\ln \left|-2 \cos (x / 2)\right| into a Fourier series (see example 6), prove that:

?
(1)

sin⁡xln⁡(2cos⁡x2)=14sin⁡x+∑n=2∞(−1)nn2−1sin⁡nx,−π<x<π\sin x \ln \left(2 \cos \frac{x}{2}\right) = \frac{1}{4} \sin x+\sum_{n = 2}^{\infty } \frac{(-1)^{n}}{n^{2}-1} \sin n x,-\pi < x < \pi;

(2)

cos⁡xln⁡(2cos⁡x2)=12−14cos⁡x+∑n=2∞(−1)nnn2−1cos⁡nx,−π<x<π\cos x \ln \left(2 \cos \frac{x}{2}\right) = \frac{1}{2}-\frac{1}{4} \cos x+\sum_{n = 2}^{\infty } \frac{(-1)^{n} n}{n^{2}-1} \cos n x,-\pi < x < \pi.

Problem 2.22.71 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove the equality for 0<x<π0 < x < \pi:

?
(1)

∑n=1∞sin⁡(2n−1)x(2n−1)3=πx8(π−x)\sum_{n = 1}^{\infty } \frac{\sin (2 n-1) x}{(2 n-1)^{3}} = \frac{\pi x}{8}(\pi -x);

(2)

∑n=2∞(−1)ncos⁡(2n−1)x(2n−3)(2n−1)(2n+1)=π8cos⁡2x−13cos⁡x\sum_{n = 2}^{\infty }(-1)^{n} \frac{\cos (2 n-1) x}{(2 n-3)(2 n-1)(2 n+1)} = \frac{\pi }{8} \cos^{2} x-\frac{1}{3} \cos x.

Problem 2.22.81 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove that

12−1π∑n=1∞sin⁡2nπxn={x−[x] for noninteger x1/2 for integer x. \frac{1}{2}-\frac{1}{\pi } \sum _{n = 1}^{\infty } \frac{\sin 2 n \pi x}{n} = \begin{cases} x-[x] & \text{ for noninteger } x \\ 1 / 2 & \text{ for integer } x. \end{cases}
?
Problem 2.22.84 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Let an,bna_{n}, b_{n} be the Fourier coefficients of the function equal to exe^{x} on the interval (−π;π)(-\pi ; \pi ). Find the sum of the series

∑n=0∞(a2n+1cos⁡(2n+1)x+b2n+1sin⁡(2n+1)x) \sum _{n = 0}^{\infty }\left(a_{2 n+1} \cos (2 n+1) x+b_{2 n+1} \sin (2 n+1) x\right)

for x∈(−π;π)x \in (-\pi ; \pi ).

?
Problem 2.22.121 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Having integrated the expansion term by term

∑n=1∞sin⁡nxn=π−x2,0<x<2π \sum _{n = 1}^{\infty } \frac{\sin n x}{n} = \frac{\pi -x}{2}, \quad 0 < x < 2 \pi

obtain the formula

∑n=1∞cos⁡nxn2=14x2−π2x+π26 \sum _{n = 1}^{\infty } \frac{\cos n x}{n^{2}} = \frac{1}{4} x^{2}-\frac{\pi }{2} x+\frac{\pi ^{2}}{6}
?
Problem 2.22.134 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove that the sequence {∑k=1nsin⁡kxk}\left\{ \sum_{k = 1}^{n} \frac{\sin k x}{k}\right\} is bounded on the entire real line.

?
Problem 2.22.146 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞cos⁡nxn!\sum_{n = 1}^{\infty } \frac{\cos n x}{n!}.

?
Problem 2.22.147 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞sin⁡nxn!\sum_{n = 1}^{\infty } \frac{\sin n x}{n!}.

?
Problem 2.22.148 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n+1cos⁡(2n−1)x(2n−1)!\sum_{n = 1}^{\infty }(-1)^{n+1} \frac{\cos (2 n-1) x}{(2 n-1)!}.

?
Problem 2.22.149 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n+1sin⁡(2n−1)x(2n−1)!\sum_{n = 1}^{\infty }(-1)^{n+1} \frac{\sin (2 n-1) x}{(2 n-1)!}.

?
Problem 2.22.150 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=0∞(−1)ncos⁡2nx(2n)!\sum_{n = 0}^{\infty }(-1)^n \frac{\cos 2 n x}{(2 n)!}.

?
Problem 2.22.151 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)nsin⁡2nx(2n)!\sum_{n = 1}^{\infty }(-1)^{n} \frac{\sin 2 n x}{(2 n) !}.

?
Problem 2.22.152 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. 1+∑n=1∞(−1)n−1cos⁡nxn(n+1)1+\sum_{n = 1}^{\infty }(-1)^{n-1} \frac{\cos n x}{n(n+1)}.

?
Problem 2.22.153 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n−1sin⁡nxn(n+1)\sum_{n = 1}^{\infty }(-1)^{n-1} \frac{\sin n x}{n(n+1)}.

?
Problem 2.22.154 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=2∞(−1)ncos⁡nxn2−1\sum_{n = 2}^{\infty }(-1)^{n} \frac{\cos n x}{n^{2}-1}.

?
Problem 2.22.155 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=2∞(−1)nsin⁡nxn2−1\sum_{n = 2}^{\infty }(-1)^{n} \frac{\sin n x}{n^{2}-1}.

?
Problem 2.22.156 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=2∞(−1)nnn2−1cos⁡nx\sum_{n = 2}^{\infty }(-1)^{n} \frac{n}{n^{2}-1} \cos n x.

?
Problem 2.22.157 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=2∞(−1)nnn2−1sin⁡nx\sum_{n = 2}^{\infty }(-1)^{n} \frac{n}{n^{2}-1} \sin n x.

?
Problem 2.22.158 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=0∞(−1)ncos⁡nx(n+1)(n+2)\sum_{n = 0}^{\infty }(-1)^{n} \frac{\cos n x}{(n+1)(n+2)}.

?
Problem 2.22.159 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n−1sin⁡nx(n+1)(n+2)\sum_{n = 1}^{\infty }(-1)^{n-1} \frac{\sin n x}{(n+1)(n+2)}.

?
Problem 2.22.160 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞sin⁡nxn(n2+a2)\sum_{n = 1}^{\infty } \frac{\sin n x}{n\left(n^{2}+a^{2}\right)}.

?
Problem 2.22.161 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)nsin⁡nxn(n2+a2)\sum_{n = 1}^{\infty }(-1)^n \frac{\sin n x}{n\left(n^2+a^2\right)}.

?
Problem 2.22.162 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n−1cos⁡(2n−1)xn\sum_{n = 1}^{\infty }(-1)^{n-1} \frac{\cos (2 n-1) x}{n}.

?
Problem 2.22.163 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. cos⁡2x1⋅2+cos⁡3x2⋅3+cos⁡4x3⋅4+…\frac{\cos 2 x}{1 \cdot 2}+\frac{\cos 3 x}{2 \cdot 3}+\frac{\cos 4 x}{3 \cdot 4}+\ldots

?
Problem 2.22.164 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. cos⁡2x1⋅2⋅3+cos⁡3x2⋅3⋅4+cos⁡4x3⋅4⋅5+…\frac{\cos 2 x}{1 \cdot 2 \cdot 3}+\frac{\cos 3 x}{2 \cdot 3 \cdot 4}+\frac{\cos 4 x}{3 \cdot 4 \cdot 5}+\ldots

?
Problem 2.22.165 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. cos⁡2x1⋅3+cos⁡4x3⋅5+cos⁡6x5⋅7+…\frac{\cos 2 x}{1 \cdot 3}+\frac{\cos 4 x}{3 \cdot 5}+\frac{\cos 6 x}{5 \cdot 7}+\ldots

?
Problem 2.22.166 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞sin⁡(2n−1)x(2n−1)3\sum_{n = 1}^{\infty } \frac{\sin (2 n-1) x}{(2 n-1)^{3}}.

?
Problem 2.22.167 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. cos⁡3x1⋅3⋅5−cos⁡5x3⋅5⋅7+cos⁡7x5⋅7⋅9+…\frac{\cos 3 x}{1 \cdot 3 \cdot 5}-\frac{\cos 5 x}{3 \cdot 5 \cdot 7}+\frac{\cos 7 x}{5 \cdot 7 \cdot 9}+\ldots

?
Problem 2.22.168 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n−1cos⁡(2n−1)x2n−1\sum_{n = 1}^{\infty }(-1)^{n-1} \frac{\cos (2 n-1) x}{2 n-1}.

?
Problem 2.22.169 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n−1sin⁡(2n−1)x2n−1\sum_{n = 1}^{\infty }(-1)^{n-1} \frac{\sin (2 n-1) x}{2 n-1}.

?
Problem 2.22.170 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n−1cos⁡(2n−1)x(2n−1)2n\sum_{n = 1}^{\infty }(-1)^{n-1} \frac{\cos (2 n-1) x}{(2 n-1) 2 n}.

?
Problem 2.22.171 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. cos⁡x+12cos⁡3x3+1⋅32⋅4cos⁡5x5+1⋅3⋅52⋅4⋅6cos⁡7x7+…\cos x+\frac{1}{2} \frac{\cos 3 x}{3}+\frac{1 \cdot 3}{2 \cdot 4} \frac{\cos 5 x}{5}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{\cos 7 x}{7}+\ldots

?
Problem 2.22.172 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. sin⁡x+12sin⁡3x3+1⋅32⋅4sin⁡5x5+…\sin x+\frac{1}{2} \frac{\sin 3 x}{3}+\frac{1 \cdot 3}{2 \cdot 4} \frac{\sin 5 x}{5}+\ldots

?
Problem 2.22.173 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. cos⁡x1⋅2+12cos⁡3x3⋅4+1⋅32⋅4cos⁡5x5⋅6+1⋅3⋅52⋅4⋅6cos⁡7x7⋅8+…\frac{\cos x}{1 \cdot 2}+\frac{1}{2} \frac{\cos 3 x}{3 \cdot 4}+\frac{1 \cdot 3}{2 \cdot 4} \frac{\cos 5 x}{5 \cdot 6}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{\cos 7 x}{7 \cdot 8}+\ldots

?
Problem 2.22.174 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. sin⁡x1⋅2+12sin⁡3x3⋅4+1⋅32⋅4cos⁡5x5⋅6+…\frac{\sin x}{1 \cdot 2}+\frac{1}{2} \frac{\sin 3 x}{3 \cdot 4}+\frac{1 \cdot 3}{2 \cdot 4} \frac{\cos 5 x}{5 \cdot 6}+\ldots

?
Problem 2.22.175 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞anncos⁡nx,∣a∣<1\sum_{n = 1}^{\infty } \frac{a^{n}}{n} \cos n x,\left|a\right| < 1.

?
Problem 2.22.176 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞sin⁡nαsin⁡nxn\sum_{n = 1}^{\infty } \frac{\sin n \alpha \sin n x}{n}.

?
Problem 2.22.177 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞sin⁡2nαsin⁡nxn,0<α<π2\sum_{n = 1}^{\infty } \frac{\sin^{2} n \alpha \sin n x}{n}, 0 < \alpha < \frac{\pi }{2}.

?
Problem 2.22.178 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Find the sum of the series. ∑n=1∞(−1)n−1rnsin⁡nxn,∣r∣⩽1\sum_{n = 1}^{\infty }(-1)^{n-1} \frac{r^{n} \sin n x}{n},\left|r\right| \leqslant 1.

?

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.