Math Problems Vault

Problem 5269

Let $S = u_{0} - u_{1} + u_{2} - u_{3} + \ldots$ be a convergent series, where the numbers $u_{0}, u_{1}, u_{2}, \ldots$ are not necessarily positive. Prove the equality:

S = \frac{u_{0}}{2} - \frac{\Delta u_{0}}{2} + \frac{\Delta u_{1}}{2} - \ldots ,

where $\Delta u_{0}, \Delta u_{1}, \Delta u_{2}, \ldots$ are the differences: $\Delta u_{0} = u_{1} - u_{0}$ , $\Delta u_{1} = u_{2} - u_{1}, \ldots$

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Problem 5270

By repeated application of the method of problem 5269, prove the equality:

S = \frac{1}{2}u_{0} - \frac{1}{4}\Delta u_{0} + \frac{1}{8}\Delta ^{2}u_{0} + \ldots + \frac{(-1)^{n-1}}{2^{n}}\Delta ^{n-1}u_{0} + (-1)^{n}R_{n},

where $2^{n}R_{n} = \Delta^{n}u_{0} - \Delta^{n}u_{1} + \Delta^{n}u_{2} - \ldots$

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Problem 5271

If the numbers $\Delta^{n}u_{0}, \Delta^{n}u_{1}, \Delta^{n}u_{2}, \ldots$ (see problem 5270) have the same sign and decrease in absolute value, then $R_{n} = \dfrac {\theta \cdot \Delta^{n}u_{0}}{2^{n}}$ ; $\dfrac {1}{2} < \theta < 1$ .

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Problem 5272

Setting $S = u_{0} - u_{1} + u_{2} - u_{3} + \ldots + (-1)^{n}u_{n} + (-1)^{n+1}\rho_{n+1}$ , where $\rho_{n+1} = u_{n+1} - u_{n+2} + u_{n+3} - \ldots$ , prove the identity

S = \frac{u_{0}}{2} - \frac{\Delta u_{0}}{2} + \frac{\Delta u_{1}}{2} - \ldots + (-1)^{m-1}\frac{\Delta u_{m-1}}{2} + (-1)^{m}\frac{\Delta \rho _{m}}{2}.

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Problem 5273

For the quantity $R_{n}$ in problem 5270, prove the equality:

2^{n}R_{n} = \Delta ^{n}\rho _{0},

where $\rho_{\nu } = u_{\nu } - u_{\nu +1} + u_{\nu +2} - \ldots$

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Problem 5274

Prove the theorem: if the series $\displaystyle \sum_{n=0}^{\infty }(-1)^{n}u_{n}$ converges, then

u_{0} - u_{1} + u_{2} - u_{3} + \ldots = \frac{1}{2}u_{0} - \frac{1}{4}\Delta u_{0} + \frac{1}{8}\Delta ^{2}u_{0} - \ldots ,

where the series on the right also converges. (Euler's transformation.)

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Problem 5275

Using the identity $a = \dfrac {a}{1+x} + \dfrac {ax}{1+x}$ , prove the identity

u_{0}x - u_{1}x^{2} + u_{2}x^{3} - \ldots = \frac{u_{0}x}{1+x} - \frac{\Delta u_{0}\cdot x^{2}}{1+x} + \frac{\Delta u_{1}\cdot x^{3}}{1+x} - \ldots ,

where the series $u_{0}x - u_{1}x^{2} + \ldots$ converges by assumption for $-1 < x < 1$ .

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Problem 5276

Under the same conditions (see problem 5275), prove the equality:

S = u_{0}y - \Delta u_{0}\cdot y^{2} + \Delta ^{2}u_{0}\cdot y^{3} - \ldots + (-1)^{n-1}\Delta ^{n-1}u_{0}\cdot y^{n} + (-1)^{n}R_{n},

where $y = \dfrac {x}{1+x}$ ; $\Delta u_{0}, \Delta^{2}u_{0}, \Delta^{3}u_{0}, \ldots$ are the successive differences, and the quantity

(1+x)^{n}R_{n} = x^{n+1}\Delta ^{n}u_{0} - x^{n+2}\Delta ^{n}u_{1} + \ldots = x\left((-1)^{n}x^{n}\rho _{0} + (-1)^{n-1}x^{n-1}\rho _{1} + \ldots + \rho _{n}\right),

where $\rho_{p} = u_{p}x^{p} - u_{p+1}x^{p+1} + \ldots$

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Problem 5277

Prove that

u_{0}x - u_{1}x^{2} + u_{2}x^{3} - \ldots = u_{0}\left(\frac{x}{1+x}\right) - \Delta u_{0}\left(\frac{x}{1+x}\right)^{2} + \Delta ^{2}u_{0}\left(\frac{x}{1+x}\right)^{3} - \ldots ,

if $0 < x < 1$ , and the series on the left converges.

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Problem 5278

Prove, for $m>0$ , $x>-\dfrac {1}{2}$ , the equality:

\frac{x}{m} - \frac{x^{2}}{m+1}\frac{x^{2}}{m+2} - \ldots = \frac{1}{m}\frac{x}{1+x} + \frac{1}{m(m+1)}\left(\frac{x}{1+x}\right)^{2} + \frac{1\cdot 2}{m(m+1)(m+2)}\left(\frac{x}{1+x}\right)^{3} + \ldots

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Problem 5279

Prove the equality:

\operatorname {arctg} x = \frac{x}{1+x^{2}}\sum _{n=0}^{\infty } \frac{2\cdot 4\cdot 6\ldots 2n}{3\cdot 5\cdot 7\ldots (2n+1)}\cdot \left(\frac{x^{2}}{x^{2}+1}\right)^{n}.

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Problem 5280

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $\ln 2 = 1 - \dfrac {1}{2} + \dfrac {1}{3} - \dfrac {1}{4} + \ldots$ (It is useful to sum the first several terms directly.)

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Problem 5281

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $\dfrac {\pi }{4} = 1 - \dfrac {1}{3} + \dfrac {1}{5} - \dfrac {1}{7} + \ldots$ (It is useful to sum the first several terms directly.)

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Problem 5282

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $1 - \dfrac {1}{3^{2}} + \dfrac {1}{5^{2}} - \dfrac {1}{7^{2}} + \ldots$ (It is useful to sum the first several terms directly.)

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Problem 5283

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $1 - \dfrac {1}{2^{3}} + \dfrac {1}{3^{3}} - \dfrac {1}{4^{3}} + \ldots$ (It is useful to sum the first several terms directly.)

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Problem 5284

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $\dfrac {1}{1\cdot 2\cdot 3} - \dfrac {1}{2\cdot 3\cdot 4} + \dfrac {1}{3\cdot 4\cdot 5} - \ldots$ (It is useful to sum the first several terms directly.)

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Problem 5285

Using Euler's transformation, find to within $10^{-5}$ the sum of the series $\dfrac {1}{\ln 10} - \dfrac {1}{\ln 11} + \dfrac {1}{\ln 12} - \dfrac {1}{\ln 13} + \ldots$ (It is useful to sum the first several terms directly.)

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Problem 5286

Prove that applying Euler's transformation to the series $\displaystyle \sum_{n=0}^{\infty }\left(-\frac{1}{2}\right)^{n}$ improves the convergence, while applying it to the series $\displaystyle \sum_{n=0}^{\infty }\left(-\frac{1}{4}\right)^{n}$ worsens it.

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Problem 5287

Setting, for the series $\sum \dfrac {1}{n^{2}}$ , $C_{n} = \dfrac {1}{n^{2} - \frac14}$ , prove the equality:

\sum _{n=1}^{\infty } \frac{1}{n^{2}} = 2 - \sum _{n=1}^{\infty } \frac{1}{n^{2}(4n^{2}-1)}.

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Remark.

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Kummer's transformation. If the series $\sum u_{n}$ converges, and the series $\sum C_{n}$ has known sum $C$ , and moreover $\dfrac {u_{n}}{C_{n}} \to \gamma$ as $n \to \infty$ , then the identity $\sum_{n=0}^{\infty } u_{n} = C\gamma + \sum_{n=0}^{\infty }(u_{n} - C_{n}\gamma )$ replaces the given series with another, more rapidly convergent one, since $\dfrac {u_{n}-C_{n}\gamma }{u_{n}} \to 0$ as $n \to \infty$ .

In particular, if one can choose a sequence of numbers $p_{1}, p_{2}, p_{3}, \ldots$ such that $p_{n}u_{n} \to 0$ and $\dfrac {p_{n}u_{n}-p_{n+1}u_{n+1}}{u_{n+1}} \to 1$ , then one may set $C_{n+1} = p_{n}u_{n} - p_{n+1}u_{n+1}$ , and then the equality

\sum _{n=1}^{\infty } u_{n} = (p_{1}+1)u_{1} + \sum _{n=1}^{\infty }(u_{n+1} - p_{n}u_{n} + p_{n+1}u_{n+1}),

holds, expressing Kummer's transformation.

Problem 5288

Prove the equality:

\sum _{n=1}^{\infty } \frac{1}{n^{2}(4n^{2}-1)} = -\frac{2}{9} - 9\sum _{n=1}^{\infty } \frac{1}{n^{2}(4n^{2}-1)(4n^{2}-9)}.

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Problem 5289

From the series $\dfrac {\pi^{2}}{6} = \displaystyle \sum_{n=1}^{\infty } \dfrac {1}{n^{2}}$ , choosing $p_{n}$ in the form $an+b+\dfrac {c}{n+d}$ , where $a,b,c,d$ are appropriate coefficients, obtain by Kummer's transformation the equality:

\pi ^{2} = 10 - \sum _{n=1}^{\infty } \frac{1}{n^{3}(n+1)^{3}}.

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Problem 5290

From the series $\dfrac {\pi^{2}}{8} = \displaystyle \sum_{n=0}^{\infty } \dfrac {1}{(2n+1)^{2}}$ , obtain by Kummer's transformation the formula:

\frac{3\pi ^{2}}{32} = 1 - 2\sum _{n=1}^{\infty } \frac{1}{(4n^{2}-1)^{3}}.

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Problem 5291

Prove the equality:

\sum _{n=1}^{\infty } \frac{1}{n^{3}} = \frac{5}{4} - \sum _{n=1}^{\infty } \frac{2n+1}{4n^{4}(n+1)^{4}}.

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Problem 5292

Prove the equality:

e = 2 + 2\sum _{n=1}^{\infty } \frac{1}{n!(n^{4}+n^{2}+1)}.

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Problem 5293

Prove the equality:

\ln 2 = \frac{3}{4} - \sum _{n=1}^{\infty } \frac{(-1)^{n-1}}{4n^{2}(n+1)^{2}}.

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Problem 5294

Prove the equality:

\sum _{n=1}^{\infty } \frac{(-1)^{n-1}}{4n^{2}(n+1)^{2}} = \frac{7}{96} - \frac{9}{2}\sum _{n=1}^{\infty } \frac{(-1)^{n-1}}{(n+1)(2n+1)(2n+3)n^{2}(n+2)^{2}}.

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Problem 5295

Prove the equality:

\frac{\pi }{4} = \sum _{n=0}^{\infty } \frac{(-1)^{n}}{2n+1} = 1 + 2\sum _{n=1}^{\infty } \frac{(-1)^{n}}{(4n^{2}-1)^{2}}.

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Problem 5296

Prove the equality:

\sum _{n=0}^{\infty } \frac{(-1)^{n}}{(2n+1)^{2}} = \frac{3}{2} + 16\sum _{n=1}^{\infty } \frac{(-1)^{n}n}{(4n^{2}-1)^{3}}.

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Problem 5297

Prove the equality:

\sum _{n=0}^{\infty } u_{n} = \frac{3}{5} - 27\sum _{n=1}^{\infty } u_{n}\cdot \frac{4n-1}{(2n-1)^{2}(8n^{2}-12n+9)(8n^{2}+4n+5)},

where $u_{n} = (-1)^{n}\left(\frac{1\cdot 3\cdot 5\ldots (2n-1)}{2\cdot 4\cdot 6\ldots 2n}\right)^{2}$ .

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Problem 5298

Kummer's transformation in its simplest form is usefully applied to trigonometric series. Applying a transformation of this type, prove, for $0<x<\pi$ , the equality:

\sum _{n=1}^{\infty } \frac{n^{2}\sin nx}{n^{3}+1} = \frac{\pi -x}{2} - \sum _{n=1}^{\infty } \frac{\sin nx}{n(n^{3}+1)}.

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Problem 5299

Applying Kummer's transformation, prove, for $0<x<\pi$ , the equality:

\sum _{n=1}^{\infty } \frac{n^{2}+1}{n^{3}+n+1}\cos nx = -\ln \left(2\sin \frac{x}{2}\right) - \sum _{n=1}^{\infty } \frac{\cos nx}{n(n^{3}+n+1)}.

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Problem 5300

Applying Kummer's transformation, prove, for $0<x<\pi$ , the equality:

\sum _{n=1}^{\infty } (-1)^{n-1}\frac{n+1}{n^{2}+1}\sin nx = \frac{1}{2}x + \sum _{n=1}^{\infty } \frac{(-1)^{n-1}(n-1)}{n(n^{2}+1)}\sin nx.

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Problem 5301

Applying Kummer's transformation, prove, for $0<x<\pi$ , the equality:

\sum _{n=1}^{\infty } \frac{n^{2}+1}{n^{4}+1}\cos nx = \frac{(\pi -x)^{2}}{4} - \frac{\pi ^{2}}{12} + \sum _{n=1}^{\infty } \frac{n^{2}-1}{n^{2}(n^{4}+1)}\cos nx.

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Problem 5302

Prove, for $0<x<\pi$ , the equality:

\sum _{n=0}^{\infty } \frac{n+1}{n^{2}+1}\sin (2n+1)x = \frac{\pi }{4} + \sum _{n=0}^{\infty } \frac{3n-1}{(n^{2}+1)(2n+1)}\sin (2n+1)x.

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Problem 5303

Prove that the function $y = \displaystyle \sum_{n=1}^{\infty } \frac{\sin nx}{n^{2}+1}$ , for $0<x<\pi$ , satisfies the differential equation

y'' - y = -\frac{1}{2}\operatorname {ctg}\frac{x}{2}.

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Problem 5304

Prove that the function $y = \displaystyle \sum_{n=1}^{\infty } \frac{n^{2}}{n^{3}+1}\sin 2nx$ , for $0<x<\dfrac {\pi }{2}$ , satisfies the equation

y^{\mathrm{VI}} + 64y = -\frac{16\cos x}{\sin ^{3}x}.

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