Section 5.9

Infinite Products

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Problem 3051

Prove the following identity:

∏n=2∞(1−1n2)=12. \prod _{n=2}^{\infty } \left(1-\frac{1}{n^{2}}\right)=\frac{1}{2}.
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Problem 3052

Prove the following identity:

∏n=2∞n3−1n3+1=23. \prod _{n=2}^{\infty } \frac{n^{3}-1}{n^{3}+1}=\frac{2}{3}.
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Problem 3053

Prove the following identity:

∏n=2∞(1−2n(n+1))=13. \prod _{n=2}^{\infty } \left(1-\frac{2}{n(n+1)}\right)=\frac{1}{3}.
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Problem 3054

Prove the following identity:

∏n=0∞[1+(12)2n]=2. \prod _{n=0}^{\infty } \left[1+\left(\frac{1}{2}\right)^{2^{n}}\right]=2.
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Problem 3055

Prove the following identity:

∏n=1∞cos⁡π2n+1=2π. \prod _{n=1}^{\infty } \cos \frac{\pi }{2^{n+1}}=\frac{2}{\pi }.
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Problem 3056

Prove the following identity:

∏n=1∞cos⁡x2n=sin⁡xx. \prod _{n=1}^{\infty } \cos \frac{x}{2^{n}}=\frac{\sin x}{x}.
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Problem 3057

Prove the following identity:

∏n=1∞ch⁡x2n=sh⁡xx. \prod _{n=1}^{\infty } \ch \frac{x}{2^{n}}=\frac{\sh x}{x}.
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Problem 3058

Prove the following identity:

∏n=0∞(1+x2n)=11−x(∣x∣<1). \prod _{n=0}^{\infty } (1+x^{2^{n}})=\frac{1}{1-x} \quad (\left|x\right|<1).
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Problem 3059

Prove the following identity:

π2=22⋅22+2⋅22+2+2⋯ . \frac{\pi }{2}=\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{2+\sqrt{2}}} \cdot \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdots .
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Problem 3060

Prove the following identity:

∏n=1∞3n3n−1⋅3n3n+1=2π33. \prod _{n=1}^{\infty } \frac{3n}{3n-1} \cdot \frac{3n}{3n+1}=\frac{2\pi }{3\sqrt{3}}.
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Problem 3061

Prove the convergence of the following infinite product and find its value:

∏n=3∞n2−4n2−1. \prod _{n=3}^{\infty } \frac{n^{2}-4}{n^{2}-1}.
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Problem 3062

Prove the convergence of the following infinite product and find its value:

∏n=1∞(1+1n(n+2)). \prod _{n=1}^{\infty } \left(1+\frac{1}{n(n+2)}\right).
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Problem 3063

Prove the convergence of the following infinite product and find its value:

∏n=1∞(2n+1)(2n+7)(2n+3)(2n+5). \prod _{n=1}^{\infty } \frac{(2n+1)(2n+7)}{(2n+3)(2n+5)}.
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Problem 3064

Prove the convergence of the following infinite product and find its value:

∏n=1∞a(−1)nn(a>0). \prod _{n=1}^{\infty } a^{\frac{(-1)^{n}}{n}} \quad (a>0).
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Problem 3065

Can the convergence of the following products be deduced from the convergence of the products ∏n=1∞pn\displaystyle \prod_{n=1}^{\infty } p_{n} and ∏n=1∞qn\displaystyle \prod_{n=1}^{\infty } q_{n}?

  1. ∏n=1∞(pn+qn)\displaystyle \prod_{n=1}^{\infty } (p_{n}+q_{n});

  2. ∏n=1∞pn2\displaystyle \prod_{n=1}^{\infty } p_{n}^{2};

  3. ∏n=1∞pnqn\displaystyle \prod_{n=1}^{\infty } p_{n}q_{n};

  4. ∏n=1∞pnqn\displaystyle \prod_{n=1}^{\infty } \frac{p_{n}}{q_{n}}.

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(a)
(b)
(c)
(d)
Problem 3066

Investigate the convergence of the following infinite product:

∏n=1∞1n. \prod _{n=1}^{\infty } \frac{1}{n}.
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Problem 3067

Investigate the convergence of the following infinite product:

∏n=1∞(n+1)2n(n+2). \prod _{n=1}^{\infty } \frac{(n+1)^{2}}{n(n+2)}.
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Problem 3068

Investigate the convergence of the following infinite product:

∏n=1∞(1+1np). \prod _{n=1}^{\infty } \left(1+\frac{1}{n^{p}}\right).
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Problem 3069

Investigate the convergence of the following infinite product:

∏n=2∞(1−1n). \prod _{n=2}^{\infty } \left(1-\frac{1}{n}\right).
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Problem 3070

Investigate the convergence of the following infinite product:

∏n=2∞(1−2n2+1)p∗). \prod _{n=2}^{\infty } \left(1-\frac{2}{n^{2}+1}\right)^{p} {}^{*)}.
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Problem 3071

Investigate the convergence of the following infinite product:

∏n=n0∞n2+a1n+b1n2+an+b, \prod _{n=n_{0}}^{\infty } \frac{n^{2}+a_{1}n+b_{1}}{n^{2}+an+b},

where n2+an+b>0n^{2}+an+b>0 for n≥n0n \geq n_{0}.

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Problem 3072

Investigate the convergence of the following infinite product:

∏n=n0∞(n−a1)(n−a2)⋯(n−ap)(n−b1)(n−b2)⋯(n−bp), \prod _{n=n_{0}}^{\infty } \frac{(n-a_{1})(n-a_{2})\cdots (n-a_{p})}{(n-b_{1})(n-b_{2})\cdots (n-b_{p})},

where n0>bin_{0}>b_{i} (i=1,2,⋯ ,pi=1,2,\cdots ,p).

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Problem 3073

Investigate the convergence of the following infinite product:

∏n=0∞n+1n+2. \prod _{n=0}^{\infty } \sqrt{\frac{n+1}{n+2}}.
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Problem 3074

Investigate the convergence of the following infinite product:

∏n=1∞nn2+1. \prod _{n=1}^{\infty } \frac{n}{\sqrt{n^{2}+1}}.
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Problem 3075

Investigate the convergence of the following infinite product:

∏n=1∞1+1nn. \prod _{n=1}^{\infty } \sqrt[n]{1+\frac{1}{n}}.
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Problem 3076

Investigate the convergence of the following infinite product:

∏n=1∞nn2. \prod _{n=1}^{\infty } \sqrt[n^{2}]{n}.
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Problem 3077

Investigate the convergence of the following infinite product:

∏n=1∞(1+xn)e−xn. \prod _{n=1}^{\infty } \left(1+\frac{x}{n}\right)e^{-\frac{x}{n}}.
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Problem 3078

Investigate the convergence of the following infinite product:

∏n=1∞(1−xc+n)exn, \prod _{n=1}^{\infty } \left(1-\frac{x}{c+n}\right)e^{\frac{x}{n}},

where c>0c>0.

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Problem 3079

Investigate the convergence of the following infinite product:

∏n=1∞(1−xn). \prod _{n=1}^{\infty } (1-x^{n}).
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Problem 3080

Investigate the convergence of the following infinite product:

∏n=1∞(1+xn2n). \prod _{n=1}^{\infty } \left(1+\frac{x^{n}}{2^{n}}\right).
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Problem 3081

Investigate the convergence of the following infinite product:

∏n=1∞[1+(1+1n)n2xn]. \prod _{n=1}^{\infty } \left[1+\frac{\left(1+\dfrac {1}{n}\right)^{n^{2}}}{x^{n}}\right].
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Problem 3082

Investigate the convergence of the following infinite product:

∏n=1∞(1−xn)exn+x22n. \prod _{n=1}^{\infty } \left(1-\frac{x}{\sqrt{n}}\right)e^{\frac{x}{\sqrt{n}}+\frac{x^{2}}{2n}}.
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Problem 3083

Investigate the convergence of the following infinite product:

∏n=1∞(1+xnnp)cos⁡xnnq. \prod _{n=1}^{\infty } \left(1+\frac{x^{n}}{n^{p}}\right)\cos \frac{x^{n}}{n^{q}}.
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Problem 3084

Investigate the convergence of the following infinite product:

∏n=1∞(sin⁡xnxn)p. \prod _{n=1}^{\infty } \left(\frac{\sin \dfrac {x}{n}}{\dfrac {x}{n}}\right)^{p}.
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Problem 3085

Investigate the convergence of the following infinite product:

∏n=1∞ln⁡(n+x)−ln⁡nn. \prod _{n=1}^{\infty } \sqrt[n]{\ln (n+x)-\ln n}.
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Problem 3086

Prove: if the series ∑n=1∞xn2\displaystyle \sum_{n=1}^{\infty } x_{n}^{2} converges, then the product ∏n=1∞cos⁡xn\displaystyle \prod_{n=1}^{\infty } \cos x_{n} converges.

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Problem 3087

Prove: if the series ∑n=1∞αn\displaystyle \sum_{n=1}^{\infty } \alpha_{n} converges absolutely, then the product ∏n=1∞tan⁡(π4+αn)\displaystyle \prod_{n=1}^{\infty } \tan \left(\frac{\pi }{4}+\alpha_{n}\right) (∣αn∣<π4\left|\alpha_{n}\right|<\dfrac {\pi }{4}) converges.

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Problem 3088

Investigate the absolute convergence and conditional convergence of the following infinite product:

∏n=1∞(1+(−1)n+1n). \prod _{n=1}^{\infty } \left(1+\frac{(-1)^{n+1}}{n}\right).
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Problem 3089

Investigate the absolute convergence and conditional convergence of the following infinite product:

∏n=1∞(1+(−1)n+1n). \prod _{n=1}^{\infty } \left(1+\frac{(-1)^{n+1}}{\sqrt{n}}\right).
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Problem 3090

Investigate the absolute convergence and conditional convergence of the following infinite product:

∏n=1∞(1+(−1)n+1np). \prod _{n=1}^{\infty } \left(1+\frac{(-1)^{n+1}}{n^{p}}\right).
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Problem 3091

Investigate the absolute and conditional convergence of the following infinite product:

∏n=2∞(1+(−1)nln⁡n). \prod _{n=2}^{\infty } \left(1+\frac{(-1)^{n}}{\ln n}\right).
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Problem 3092

Investigate the absolute and conditional convergence of the following infinite product:

∏n=2∞nn+(−1)n. \prod _{n=2}^{\infty } \frac{\sqrt{n}}{\sqrt{n}+(-1)^{n}}.
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Problem 3093

Investigate the absolute and conditional convergence of the following infinite product:

∏n=1∞n(−1)n. \prod _{n=1}^{\infty } n^{(-1)^{n}}.
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Problem 3094

Investigate the absolute and conditional convergence of the following infinite product:

∏n=1∞n(−1)nn. \prod _{n=1}^{\infty } \sqrt[n]{n^{(-1)^{n}}}.
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Problem 3095

Investigate the absolute and conditional convergence of the following infinite product:

∏n=1∞[1+(−1)n(n−1)2n]. \prod _{n=1}^{\infty } \left[1+\frac{(-1)^{\frac{n(n-1)}{2}}}{n}\right].
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Problem 3096

Investigate the absolute and conditional convergence of the following infinite product:

(1+11)(1−13)(1−15)×(1+12)(1−17)(1−19)(1+13)⋯ . \begin{align} & \left(1+\frac{1}{\sqrt{1}}\right)\left(1-\frac{1}{\sqrt{3}}\right)\left(1-\frac{1}{\sqrt{5}}\right) \\ & \quad \times \left(1+\frac{1}{\sqrt{2}}\right)\left(1-\frac{1}{\sqrt{7}}\right)\left(1-\frac{1}{\sqrt{9}}\right)\left(1+\frac{1}{\sqrt{3}}\right)\cdots . \end{align}
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Problem 3097

Investigate the absolute and conditional convergence of the following infinite product:

(1+11a)(1−12a)2(1+13a)(1+14a)(1−15a)2×(1+16a)⋯ . \begin{align} & \left(1+\frac{1}{1^{a}}\right)\left(1-\frac{1}{2^{a}}\right)^{2}\left(1+\frac{1}{3^{a}}\right)\left(1+\frac{1}{4^{a}}\right)\left(1-\frac{1}{5^{a}}\right)^{2} \\ & \quad \times \left(1+\frac{1}{6^{a}}\right)\cdots . \end{align}
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Problem 3098

Prove that although the series

(12+12)+(−12)+(13+13)+(−13)+⋯(1) \left(\frac{1}{\sqrt{2}}+\frac{1}{2}\right)+\left(-\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{\sqrt{3}}+\frac{1}{3}\right)+\left(-\frac{1}{\sqrt{3}}\right)+\cdots \tag {1}

diverges, the product

(1+12+12)(1−12)(1+13+13)(1−13)⋯(2) \left(1+\frac{1}{\sqrt{2}}+\frac{1}{2}\right)\left(1-\frac{1}{\sqrt{2}}\right)\left(1+\frac{1}{\sqrt{3}}+\frac{1}{3}\right)\left(1-\frac{1}{\sqrt{3}}\right)\cdots \tag {2}

converges.

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Problem 3099

Prove that although the two series

∑n=1∞αnand∑n=1∞αn2 \sum _{n=1}^{\infty } \alpha _{n} \quad \text{and} \quad \sum _{n=1}^{\infty } \alpha _{n}^{2}

diverge, the product ∏n=1∞(1+αn)\displaystyle \prod_{n=1}^{\infty } (1+\alpha_{n}) converges, where

αn={−1k,if n=2k−1,1k+1k+1kk,if n=2k. \alpha _{n}= \begin{cases} -\dfrac {1}{\sqrt{k}}, & \text{if } n=2k-1, \\ \dfrac {1}{\sqrt{k}}+\dfrac {1}{k}+\dfrac {1}{k\sqrt{k}}, & \text{if } n=2k. \end{cases}
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Problem 3100

Let

ζ(x)=∑n=1∞1nx \zeta (x)=\sum _{n=1}^{\infty } \frac{1}{n^{x}}

(the Riemann ζ\zeta function) and let pnp_{n} (n=1,2,⋯n=1,2,\cdots) be the sequence of primes. Prove:

∏n=1∞(1−1pnx)−1=ζ(x). \prod _{n=1}^{\infty } \left(1-\frac{1}{p_{n}^{x}}\right)^{-1}=\zeta (x).
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Problem 3101

Prove that the product

∏n=1∞(1−1pn)−1 \prod _{n=1}^{\infty } \left(1-\frac{1}{p_{n}}\right)^{-1}

and the series

∑n=1∞1pn \sum _{n=1}^{\infty } \frac{1}{p_{n}}

(where pnp_{n}, n=1,2,⋯n=1,2,\cdots, is the sequence of primes) diverge (Euler).

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Problem 3102

Let an>0a_{n}>0 (n=1,2,⋯n=1,2,\cdots), and

anan+1=1+pn+O(1n1+C)(C>0). \frac{a_{n}}{a_{n+1}}=1+\frac{p}{n}+O\left(\frac{1}{n^{1+C}}\right) \quad (C>0).

Prove that an=O∗(1np)a_{n}=O^{*}\left(\dfrac {1}{n^{p}}\right).

Hint: investigate lim⁡n→∞annp=a1∏n=1∞an+1an(1+1n)p\displaystyle \lim_{n \rightarrow \infty } a_{n}n^{p}=a_{1}\prod_{n=1}^{\infty } \dfrac {a_{n+1}}{a_{n}}\left(1+\dfrac {1}{n}\right)^{p}.

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Problem 3103

Use Wallis's formula to prove

1⋅3⋅5⋯(2n−1)2⋅4⋅6⋯(2n)∼1πn. \frac{1 \cdot 3 \cdot 5\cdots (2n-1)}{2 \cdot 4 \cdot 6\cdots (2n)} \sim \frac{1}{\sqrt{\pi n}}.
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Problem 3104

Prove that the expression

an=n! ennn+12 a_{n}=\frac{n! \, e^{n}}{n^{n+\frac{1}{2}}}

has a nonzero limit AA as n→∞n \rightarrow \infty.

Deduce from this Stirling's formula

n!=Ann+12e−n(1+εn), n!=An^{n+\frac{1}{2}}e^{-n}(1+\varepsilon _{n}),

where lim⁡n→∞εn=0\displaystyle \lim_{n \rightarrow \infty } \varepsilon_{n}=0 and A=2πA=\sqrt{2\pi }.

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Problem 3105

According to Euler's definition, the gamma function Γ(x)\Gamma (x) is determined by the following formula:

Γ(x)=lim⁡n→∞n! nxx(x+1)⋯(x+n). \Gamma (x)=\lim _{n \rightarrow \infty } \frac{n! \, n^{x}}{x(x+1)\cdots (x+n)}.

Starting from this formula:

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(a)

Express the function Γ(x)\Gamma (x) in the form of an infinite product;

(b)

Prove that Γ(x)\Gamma (x) is meaningful for all real xx that are not negative integers;

(c)

Deduce the following property

Γ(x+1)=xΓ(x); \Gamma (x+1)=x\Gamma (x);
(d)

Find the value of Γ(n)\Gamma (n) for positive integer nn.

Problem 3106

Let the function f(x)f(x) be integrable on the closed interval [a,b][a,b], and

δn=b−an,fin=f(a+iδn)(i=1,2,⋯ ,n), \delta _{n}=\frac{b-a}{n},\quad f_{in}=f(a+i\delta _{n}) \quad (i=1,2,\cdots ,n),

Prove

lim⁡n→∞∏i=1n(1+δnfin)=e∫abf(x) dx. \lim _{n \rightarrow \infty } \prod _{i=1}^{n} (1+\delta _{n}f_{in})=e^{\int _{a}^{b} f(x)\, dx}.
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Problem 3107

Prove that:

lim⁡n→∞∏i=0n−1(a+ib)n1n∑i=0n−1(a+ib)=2e, \lim _{n \rightarrow \infty } \frac{\sqrt[n]{\displaystyle \prod _{i=0}^{n-1} (a+ib)}}{\dfrac {1}{n}\displaystyle \sum _{i=0}^{n-1} (a+ib)}=\frac{2}{e},

where a>0a>0 and b>0b>0.

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Problem 3108

Let fn(x)f_{n}(x) (n=1,2,⋯n=1,2,\cdots) be continuous functions on the interval (a,b)(a,b), and

∣fn(x)∣≤cn(n=1,2,⋯ ), \left|f_{n}(x)\right| \leq c_{n} \quad (n=1,2,\cdots ),

where the series ∑n=1∞cn\displaystyle \sum_{n=1}^{\infty } c_{n} converges. Prove that the function

F(x)=∏n=1∞[1+fn(x)](∣fn(x)∣<1) F(x)=\prod _{n=1}^{\infty } \left[1+f_{n}(x)\right] \quad (\left|f_{n}(x)\right|<1)

is continuous on the interval (a,b)(a,b).

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Problem 3109

Find an expression for the derivative of the function

F(x)=∏n=1∞[1+fn(x)] F(x)=\prod _{n=1}^{\infty } \left[1+f_{n}(x)\right]

What are sufficient conditions for F′(x)F'(x) to exist?

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Problem 3110

Prove that if 0<x<y0<x<y, then

lim⁡n→∞x(x+1)⋯(x+n)y(y+1)⋯(y+n)=0. \lim _{n \rightarrow \infty } \frac{x(x+1)\cdots (x+n)}{y(y+1)\cdots (y+n)}=0.
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