Summing numerical series via Fourier expansions

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Problem 1.7.2 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 1.7 "Fourier Series"

Let f:R→Rf : \mathbb {R} \rightarrow \mathbb {R} be the function of period 2π2\pi such that f(x)=x3f(x) = x^{3} for −π≤x<π-\pi \leq x < \pi.

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(1)

Prove that the Fourier series for ff has the form ∑1∞bnsin⁡nx\sum_{1}^{\infty } b_{n}\sin nx and write an integral formula for bnb_{n} (do not evaluate it).

(2)

Prove that the Fourier series converges for all xx.

(3)

Prove

∑n=1∞bn2=2π67. \sum _{n=1}^{\infty } b_{n}^{2} = \frac{2\pi ^{6}}{7}.
Problem 3.70 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 3.3 "Orthogonal and orthonormal systems and bases in Hilbert spaces"

Using Parseval's equality, compute the sums of the series:

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(a)

∑n=1∞1n2\sum_{n = 1}^{\infty } \frac{1}{n^{2}};

(b)

∑n=1∞1n4\sum_{n = 1}^{\infty } \frac{1}{n^{4}}.

Problem 2938 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.6 "Fourier Series"

Expand the function f(x)=sgn⁡xf(x)=\operatorname {sgn}x (−π<x<π-\pi <x<\pi) into a Fourier series. Sketch the graph of the function and the graphs of several partial sums of its Fourier series. Using the expansion, find the sum of the Leibniz series ∑n=1∞(−1)n−12n−1\displaystyle \sum_{n=1}^{\infty } \dfrac {(-1)^{n-1}}{2n-1}.

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Problem 1.48 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 1 "Banach Spaces and Operator Theory"

Let ψ=χ[0,1/2)−χ[1/2,1)\psi = \chi_{[0,1 / 2)}-\chi_{[1 / 2,1)} be the Haar wavelet. Compute the Fourier coefficients of ψ\psi and apply the Plancherel Equality to show that π28=∑n=1∞1(2n−1)2\frac{\pi^{2}}{8} = \sum_{n = 1}^{\infty } \frac{1}{(2 n-1)^{2}}.

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Problem 13.25 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

Let f(x)=π2(x2−x+16)f(x) = \pi^{2}\left(x^{2}-x+\frac{1}{6}\right) for x∈[0,1)x \in [0,1). When extended 1 -periodically, ff is a continuous function on T\mathbf{T}. Compute f^\widehat{f} and show that f^∈ℓ1(Z)\widehat{f} \in \ell^{1}(\mathbb {Z}). Use this to show that

∑n=1∞cos⁡2πnxn2=π2(x2−x+16),x∈[0,1](13.20) \sum _{n = 1}^{\infty } \frac{\cos 2 \pi n x}{n^{2}} = \pi ^{2}\left(x^{2}-x+\frac{1}{6}\right), \quad x \in [0,1] \tag {13.20}

where the series converges uniformly on [0,1][0,1]. What does the series converge to for other xx? Take x=0x = 0 to obtain Euler's formula:

∑n=1∞1n2=π26 \sum _{n = 1}^{\infty } \frac{1}{n^{2}} = \frac{\pi ^{2}}{6}

Compare Exercise 1.55.

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Problem 2.22.12 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 f(x)=sign⁡x,−π<x<πf(x) = \operatorname {sign} x,-\pi < x < \pi, into a Fourier series and, using the resulting expansion, find the sum of the Leibniz series

∑n=0∞(−1)n2n+1 \sum _{n = 0}^{\infty } \frac{(-1)^{n}}{2 n+1}
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Problem IX.4.4 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section IX.4 "Fourier Series and Transform"

Using the cosine Fourier coefficients of an even function, give an example of a nonzero series ∑an\sum a_{n} such that ∑j⩾1akj=0\sum_{j \geqslant 1} a_{k j}=0 for any k∈Nk \in \mathbb {N} (cf. Problem IV.3.8).

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Problem 1.8.12 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.8 "Some Special Functions"

Suppose 0<δ<π,f(x)=10 < \delta < \pi , f(x) = 1 if ∣x∣≤δ,f(x)=0\left|x\right| \leq \delta , f(x) = 0 if δ<∣x∣≤π\delta < \left|x\right| \leq \pi, and f(x+2π)=f(x)f(x+2 \pi ) = f(x) for all xx.

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(a)

Compute the Fourier coefficients of ff.

(b)

Conclude that

∑n=1∞sin⁡(nδ)n=π−δ2(0<δ<π) \sum _{n = 1}^{\infty } \frac{\sin (n \delta )}{n} = \frac{\pi -\delta }{2} \quad (0 < \delta < \pi )
(c)

Deduce from Parseval's theorem that

∑n=1∞sin⁡2(nδ)n2δ=π−δ2 \sum _{n = 1}^{\infty } \frac{\sin ^{2}(n \delta )}{n^{2} \delta } = \frac{\pi -\delta }{2}
(d)

Let δ→0\delta \rightarrow 0 and prove that

∫0∞(sin⁡xx)2dx=π2 \int _{0}^{\infty }\left(\frac{\sin x}{x}\right)^{2} d x = \frac{\pi }{2}
(e)

Put δ=π/2\delta = \pi / 2 in (c). What do you get?

Problem 1.8.14 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.8 "Some Special Functions"

If f(x)=(π−∣x∣)2f(x) = (\pi -\left|x\right|)^{2} on [−π,π][-\pi , \pi ], prove that

f(x)=π23+∑n=1∞4n2cos⁡nx f(x) = \frac{\pi ^{2}}{3}+\sum _{n = 1}^{\infty } \frac{4}{n^{2}} \cos n x

and deduce that

∑n=1∞1n2=π26,∑n=1∞1n4=π490 \sum _{n = 1}^{\infty } \frac{1}{n^{2}} = \frac{\pi ^{2}}{6}, \quad \sum _{n = 1}^{\infty } \frac{1}{n^{4}} = \frac{\pi ^{4}}{90}

(A recent article by E. L. Stark contains many references to series of the form ∑n−s\sum n^{-s}, where ss is a positive integer. See Math. Mag., vol. 47, 1974, pp. 197-202).

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Problem 4.2.5 from Saxe K. "Beginning Functional Analysis" [Sax02]More problems in Section 4.2 "Bessel's Inequality, Parseval's Theorem, and the Riesz-Fischer Theorem"
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(a)

Show that the classical Fourier series of f(x)=xf(x)=x is

2∑n=1∞(−1)n+1nsin⁡(nx) 2 \sum _{n=1}^{\infty } \frac{(-1)^{n+1}}{n} \sin (n x)
(b)

Use your work in (a), together with Parseval's identity, to obtain Euler's remarkable identity

∑n=1∞1n2=π26 \sum _{n=1}^{\infty } \frac{1}{n^{2}}=\frac{\pi ^{2}}{6}

(Note: The same procedure can be applied to the Fourier series of x2x^{2} to obtain

∑n=1∞1n4=π490 \sum _{n=1}^{\infty } \frac{1}{n^{4}}=\frac{\pi ^{4}}{90}

and so on!)

Problem 26.28 from Billingsley P. "Probability and Measure" [Bil95]More problems in Section 26 "Characteristic Functions"

↑\uparrow Show that

(π−x)2=π23+4∑m=1∞cos⁡mxm2,0≤x≤2π (\pi -x)^{2}=\frac{\pi ^{2}}{3}+4 \sum _{m=1}^{\infty } \frac{\cos m x}{m^{2}}, \quad 0 \leq x \leq 2 \pi

Show that ∑m=1∞1/m2=π2/6\sum_{m=1}^{\infty } 1 / m^{2}=\pi^{2} / 6 and ∑m=1∞(−1)m+1/m2=π2/12\sum_{m=1}^{\infty }(-1)^{m+1} / m^{2}=\pi^{2} / 12.

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References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.
[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.
[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[Hei11] Heil C. A Basis Theory Primer. Birkhäuser, 2011.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.
[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.
[Rud76] Rudin W. Principles of Mathematical Analysis. 3rd ed., McGraw-Hill, 1976.
[Sax02] Saxe K. Beginning Functional Analysis. Springer, 2002.
[Bil95] Billingsley P. Probability and Measure. 3rd ed., Wiley-Interscience, 1995.