Miscellaneous Problems

Prove Catalan's identity

and, using it, show that $\sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{n} = \ln 2$.

Prove the equalities:

where $C$ is Euler's constant, and $\varepsilon_{n}$ and $\eta_{n} \rightarrow 0$ as $n \rightarrow \infty$.

Prove the equalities:

Let the series $\sum u_{n}$ be obtained from the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\ldots$ by a rearrangement of terms such that the terms of each sign decrease in absolute value, and among the first $n$ terms the ratio of the number of positive terms to the number of negative terms tends to $\sigma$ as $n \rightarrow \infty$. Prove the equality: $\sum_{\nu =1}^{\infty } u_{\nu } = \ln \sqrt{4\sigma }$.

Prove the equalities:

Prove the equality $1+\frac{1}{5}-\frac{1}{3}+\frac{1}{9}+\frac{1}{13}-\frac{1}{7}+\ldots = \frac{\pi }{4}+\frac{1}{4}\ln 2$.

Prove the equality $1-\frac{1}{3}-\frac{1}{7}+\frac{1}{5}-\frac{1}{11}-\frac{1}{15}+\frac{1}{9}-\ldots = \frac{\pi }{4}-\frac{1}{4}\ln 2$.

From the harmonic series $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\ldots$, remove the terms whose denominator contains the digit $9$. Prove that the resulting series converges, and that its sum is less than $80$.

Prove the equality $s_{2}+s_{3}+s_{4}+\ldots = 1$; $s_{\nu } = \sum_{n=2}^{\infty } \frac{1}{n^{\nu }}$.

Prove the equality $s_{3}+s_{5}+s_{7}+\ldots = \frac{1}{4}$.

Prove the equality $s_{2}+\frac{1}{2}s_{4}+\frac{1}{3}s_{6}+\ldots = \ln 2$.

Let $a_{n}$ be the numbers of the natural sequence of the form $p^{q}$, where $p$ and $q$ are integers greater than one. Prove Euler's equality: $\sum_{n=1}^{\infty } \frac{1}{a_{n}-1} = 1$.

Let $a_{n}$ be the numbers of the natural sequence of the form $p^{q}$, where $p$ and $q$ are integers greater than one. Prove Euler's equality: $\sum_{n=1}^{\infty } \frac{1}{a_{n}+1} = \frac{\pi^{2}}{3}-\frac{5}{2}$.

Let $a_{n}$ be the numbers of the natural sequence of the form $p^{q}$, where $p$ and $q$ are integers greater than one, and let $b_{m}$ be those of the numbers $a_{n}$ that are divisible by $2$. Prove Euler's equality: $\sum_{n=1}^{\infty } \frac{1}{b_{m}-1} = \ln 2$.

Let $a_{n}$ be the numbers of the natural sequence of the form $p^{q}$, where $p$ and $q$ are integers greater than one, and let $b_{m}$ be those of the numbers $a_{n}$ that are divisible by $2$. Prove Euler's equality: $\sum_{n=1}^{\infty } \frac{1}{b_{m}+1} = \frac{\pi^{2}}{12}+\ln 2-1$.

Prove the equality $\varphi (x)+\varphi (1-x) = \frac{\pi^{2}}{6}-\ln x\ln (1-x)$; $\varphi (x) = \sum_{n=1}^{\infty } \frac{x^{n}}{n^{2}}$.

Prove the equality $\varphi (1-x)+\varphi \left(\frac{x-1}{x}\right) = -\frac{1}{2}(\ln x)^{2}$, where $\varphi (x) = \sum_{n=1}^{\infty } \frac{x^{n}}{n^{2}}$ (the same function as in problem 5141).

Prove the equality $\sum_{n=1}^{\infty } \frac{1}{n^{2}\cdot 2^{n}} = \frac{\pi^{2}}{12}-\frac{1}{2}(\ln 2)^{2}$.

Prove the equality $\sum_{n=1}^{\infty } \frac{(-2)^{n}s_{n}}{n!} = 0$; $s_{n} = 1^{3}+2^{3}+3^{3}+\ldots +n^{3}$.

Prove the equality $\sum_{n=0}^{\infty } \frac{n^{m}}{n!} = e\cdot p_{m}$, where the integers $p_{m}$ are related by the equality:

Prove the equality

here $\sigma > n$, $0 < p < 1$, $p+q = 1$.

Prove that the sum of the series

equals $\frac{a_{1}}{x}$, if the series $\sum \frac{1}{a_{n}}$ is divergent, and equals

if the series $\sum \frac{1}{a_{n}}$ is convergent. The numbers $x, a_{1}, a_{2}, a_{3}, \ldots$ are positive.

Prove the equality $\sum_{n=2}^{\infty } \frac{n}{(1+2x)(1+3x)\ldots (1+nx)} = \frac{1}{x}$; $x > 0$.

Prove the equality $\sum_{n=2}^{\infty } \frac{((n-1)!)^{2}}{(x^{2}+2^{2})(x^{2}+3^{2})\ldots (x^{2}+n^{2})} = \frac{1}{x^{2}}\left(1-\frac{\pi x(x^{2}+1)}{\operatorname {sh}\pi x}\right)$.

Find the sum of the series $\sum_{n=1}^{\infty } \frac{1}{a^{2}+n^{2}}$.

Find the sum of the series $\sum_{n=1}^{\infty } \frac{1}{(a^{2}+n^{2})^{2}}$.

Find the sum of the series $\sum_{-\infty }^{\infty } \frac{1}{x^{4}+n^{4}}$.

Prove the following statement: $\lim_{x \rightarrow \infty } \sum_{n=1}^{\infty } \frac{x^{2}}{x^{2}+n^{2}} = \frac{\pi }{2}$.

Prove the following statement: if $u_{n} > 0$, $u_{n}-u_{n+1} > 0$, $u_{n}-2u_{n+1}+u_{n+2} > 0$, then

Prove the following statement: $\lim_{x \rightarrow 1}\left(\frac{x}{1+x}-\frac{x^{2}}{1+x^{2}}+\frac{x^{3}}{1+x^{3}}-\ldots \right) = \frac{1}{4}$.

Prove the following statement: if $a_{n} > 0$, $b_{n} > 0$, the series $\sum a_{n}x^{n}$ is divergent at $x = 1$, and convergent for $\left|x\right| < 1$, and $\frac{a_{n}}{b_{n}} \rightarrow A$ as $n \rightarrow \infty$, then

Prove the following statement: if $\sum a_{n}$ diverges and $\frac{A_{n}}{B_{n}} \rightarrow A$, where $A_{n} = \sum_{\nu =1}^{n} a_{\nu }$, $B_{n} = \sum_{\nu =1}^{n} b_{\nu }$, then

Prove the following statement: $\lim_{x \rightarrow 1-0} (1-x)^{p}\sum_{n=1}^{\infty } n^{p-1}x^{n} = \Gamma (p)$; $p > 0$.

Prove the following statement: if $\frac{a_{n}}{n^{p-1}} \rightarrow a$ as $n \rightarrow \infty$, then

Prove the following statement: if $\lim_{n \rightarrow \infty } \frac{a_{1}+a_{2}+\ldots +a_{n}}{n^{p}} = a$, then

Prove the following statement: $\lim_{x \rightarrow 1-0} \sqrt{1-x}\, (x+x^{4}+x^{9}+\ldots ) = \frac{1}{2}\sqrt{\pi }$.

Prove Poisson's formula:

Prove the equality $\frac{1}{2}+\sum_{n=1}^{\infty } e^{-\pi n^{2}x} = \frac{1}{\sqrt{x}}\left(\frac{1}{2}+\sum_{n=1}^{\infty } e^{-\frac{\pi n^{2}}{x}}\right)$.

Prove the following statement: $\lim_{x \rightarrow 1-0} (x-x^{4}+x^{9}-x^{16}+\ldots ) = \frac{1}{2}$.

Prove the equality $\sum_{n=-\infty }^{\infty } e^{-\pi a(n+x)^{2}} = \frac{1}{\sqrt{a}}\sum_{n=-\infty }^{\infty } e^{-\frac{\pi n^{2}}{a}}\cos 2\pi nx$.

Prove the following statement (Borel's theorem): if the series $\sum a_{n}$ converges and has sum $s$, and moreover

Prove the following statement: if $R_{1}$ and $R_{2}$ are the radii of convergence of the series $\sum a_{n}x^{n}$ and $\sum b_{n}x^{n}$, then the radius of convergence of the series $\sum a_{n}b_{n}x^{n}$ is no less than $R_{1}R_{2}$.

Prove the following statement: for the convergence of a series $\sum a_{n}$ with positive decreasing terms, the condition $\lim_{n \rightarrow \infty } na_{n} = 0$ is necessary.

Prove the following statement: if $f(x) < M$ for $0 < x < 2\pi$, then for its Fourier coefficients the inequalities

hold.

Prove the following statement: the series $\sum \frac{\cos x\sqrt{n}}{n^{\sigma }}$ and $\sum \frac{\sin x\sqrt{n}}{n^{\sigma }}$ for $x \neq 0$ converge for $\sigma > \frac{1}{2}$ and diverge for $\sigma < \frac{1}{2}$.

Prove the following statement: the series $\sum \frac{(-1)^{\left[\sqrt{n}\right]}}{n^{\sigma }}$, where $\left[\sqrt{n}\right]$ is the integer part of $\sqrt{n}$, converges for $\sigma > \frac{1}{2}$ and diverges for $\sigma < \frac{1}{2}$.

Using the equality $\frac{1}{a} = \int_{0}^{1} x^{a-1}\, dx$, $a > 0$, prove the formula:

Prove the identity:

Prove the equality for $\left|q\right| < 1$:

Prove the equality for $\left|q\right| < 1$:

Prove the equality for $\left|q\right| < 1$:

Using the preceding result (problem 5176), obtain the Jacobi identity:

By a change of variables from the preceding result (problem 5177), obtain Euler's identity:

Prove the Jacobi identities (for $\left|q\right| < 1$):

where $q_{0} = \prod_{n=1}^{\infty } (1-q^{2n})$.

The Jacobi theta functions are defined by the equalities (for $\left|q\right| < 1$):

Prove the equalities (for $\left|q\right| < 1$):

where $q_{0}$ has the same value as before (see problem 5179): $q_{0} = \prod_{n=1}^{\infty } (1-q^{2n})$.

Setting

and $f(q) = ABC$, where $\left|q\right| < 1$, prove that $f(q) = f(q^{2})$ and, consequently, $ABC = 1$.

Prove Jacobi's equality: $\vartheta_{1}' = \pi \vartheta_{2}\vartheta_{3}\vartheta_{4}$, where $\vartheta_{2}, \vartheta_{3}, \vartheta_{4}$, $\vartheta_{1}'$ are the values of the theta functions and of their derivative at $v = 0$ (see problem 5180 for the definition of the theta functions).

Prove the equality $\sqrt{N} = a\left(1+\beta +\frac{1\cdot 3}{2!}\beta^{2}+\frac{1\cdot 3\cdot 5}{3!}\beta^{3}+\ldots \right)$, where $N = a^{2}+b$; $\beta = \frac{b}{2N}$; $\left|b\right| < N$.

Prove the equality $\sqrt[3]{N} = a\left(1+\beta +\frac{1\cdot 4}{2!}\beta^{2}+\frac{1\cdot 4\cdot 7}{3!}\beta^{3}+\ldots \right)$, where $N = a^{3}+b$; $\beta = \frac{b}{3N}$; $\left|b\right| < N$.

Verify the equality $\sqrt{2} = 1{,}41421\ 35623\ 73095\ 04880\ 16885\ldots$

Verify the equality $\sqrt[3]{2} = 1{,}25991\ 20498\ 94873\ 16476\ 72106\ldots$

Verify Ramanujan's equality: