Math Problems Vault

Problem 1646

(a)(b)

By direct division, arranging the polynomials in ascending powers of the variable $x$ , prove the series expansions:

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(a)

$\dfrac {1}{a-bx} = \dfrac {1}{a}+\dfrac {bx}{a^{2}}+\dfrac {b^{2}x^{2}}{a^{3}}+\ldots$ ;

(b)

$\dfrac {1}{a+bx} = \dfrac {1}{a}-\dfrac {bx}{a^{2}}+\dfrac {b^{2}x^{2}}{a^{3}}-\ldots$ , $\left(\left|x\right| < \left|\dfrac {a}{b}\right|\right)$ .

Problem 1647

Using the identity $\dfrac {5-2x}{6-5x+x^{2}} = \dfrac {1}{2-x}+\dfrac {1}{3-x}$ , prove the series expansion:

\frac{5-2x}{6-5x+x^{2}} = \frac{5}{6}+\frac{13}{36}x+\frac{35}{216}x^{2}+\ldots +\left(\frac{1}{2^{n}}+\frac{1}{3^{n}}\right)x^{n-1}+\ldots , \quad \left|x\right| < 2.

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Problem 1648

Assuming the series expansion

\frac{1}{1-x-x^{2}} = a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots

to be legitimate, and multiplying both sides of the equality by $1-x-x^{2}$ , prove that $a_{0}=1$ , $a_{1}=1$ , $a_{2}=2$ , $a_{3}=3$ , $a_{4}=5, \ldots$ and in general that $a_{0}, a_{1}, a_{2}, \ldots$ are the Fibonacci numbers, defined by the equalities $a_{n}=a_{n-1}+a_{n-2}$ for $n \geq 2$ , $a_{0}=1$ , $a_{1}=1$ .

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Problem 1649

Using the identity

\frac{1}{1-x-x^{2}} = \frac{1}{\sqrt{5}}\left(\frac{1}{\frac{\sqrt{5}+1}{2}+x}+\frac{1}{\frac{\sqrt{5}-1}{2}-x}\right),

find a formula giving the general term $a_{n}$ of the Fibonacci series.

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Problem 1650

Prove that in the series expansion of the fraction

\frac{a+bx+cx^{2}}{A+Bx+Cx^{2}+Dx^{3}} = a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots ,

which holds for $A \neq 0$ and for sufficiently small values of $\left|x\right|$ , there exists the relation between the coefficients $a_{n}$ : $Aa_{n}+Ba_{n-1}+Ca_{n-2}+Da_{n-3}=0$ ; $n > 2$ .

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Problem 1651

Expand the polynomial $x^{4}-5x^{3}+5x^{2}+x+2$ in powers of $x-2$ .

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Problem 1652

Expand the polynomial $x^{5}+2x^{4}-x^{2}+x+1$ in powers of $x+1$ .

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Problem 1653

Prove the equalities:

\sin (a+x) = \sin a \cos +\cos a \sin x,

\cos (a+x) = \cos a \cos x-\sin a \sin x,

by expanding the left-hand sides in powers of $x$ .

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Problem 1654

In the expansion for $\ln (1+x)$ , replace $x$ by $-x$ ; starting from the series for $\ln (1+x)$ and $\ln (1-x)$ , obtain Gregory's series:

\ln \frac{1+x}{1-x} = 2\left(x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\ldots \right), \quad \left|x\right| < 1.

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Problem 1655

Obtain the equality

\ln (N+1)-\ln N = 2\left(\frac{1}{2N+1}+\frac{1}{3(2N+1)^{3}}+\frac{1}{5(2N+1)^{5}}+\ldots \right), \quad N > 0.

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Problem 1656

From the series for $e^{x}$ obtain the expansion for $e^{-x}$ , as well as the expansions:

\ch x \equiv \frac{e^{x}+e^{-x}}{2} = 1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\ldots ,

\sh x \equiv \frac{e^{x}-e^{-x}}{2} = x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\ldots ,

valid for every $x$ .

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Problem 1657

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $1+\cos x+\cos 2x+\ldots +\cos (n-1)x$ .

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Problem 1658

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $\cos x+\cos 3x+\cos 5x+\ldots +\cos (2n-1)x$ .

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Problem 1659

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $\sin x+\sin 2x+\sin 3x+\ldots +\sin nx$ .

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Problem 1660

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $\sin x-\sin 2x+\ldots -\sin 2nx$ .

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Problem 1661

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $\sin^{2}x-\sin^{2}3x+\sin^{2}5x-\ldots +(-1)^{n-1}\sin^{2}(2n-1)x$ .

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Problem 1662

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $x\sin a+x^{2}\sin 2a+\ldots +x^{n-1}\sin (n-1)x$ .

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Problem 1663

Prove the equality

\sum _{n=0}^{\infty } \frac{x^{4n}}{(4n)!} = \frac{1}{4}(e^{x}+e^{-x}+2\cos x).

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Problem 1664

Prove the equality

\sum _{n=0}^{\infty } \frac{x^{3n}}{(3n)!} = \frac{1}{3}\left(e^{x}+2e^{-x/2}\cos \frac{x\sqrt{3}}{2}\right).

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Problem 1665

Prove the equality

e^{x}\cos x = \sum _{n=0}^{\infty } \frac{2^{n/2}\cos \frac{\pi n}{4}}{n!}x^{n}.

Here, as elsewhere, it is assumed that $0! = 1$ .

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Problem 1666

Prove the equality

\ch x \cdot \cos x = \sum _{n=0}^{\infty } (-1)^{n}\frac{2^{2n}x^{4n}}{(4n)!}.

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Problem 1667

Prove the equality

e^{x\ctg a}\cos x = \sum _{n=0}^{\infty } \frac{\cos na}{\sin ^{n}a} \cdot \frac{x^{n}}{n!}.

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Problem 1668

Prove the equality

(1+x^{2})^{m/2}\sin (m\arctg x) = \sum _{\nu =0}^{\infty } (-1)^{\nu }\frac{m(m-1)\ldots (m-2\nu )}{(2\nu +1)!}x^{2\nu +1}, \quad \left|x\right| < 1.

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Problem 1669

Prove the equality

(1+x^{2})^{-m/2}\cos (m\arctg x) = \sum _{\nu =0}^{\infty } (-1)^{\nu }\frac{m(m+1)\ldots (m+2\nu -1)}{(2\nu )!}x^{2\nu }, \quad \left|x\right| < 1.

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Problem 1670

Applying Lagrange's formula to $\varphi (x)-\varphi (\pi )$ , where

\varphi (x) = \frac{x}{2}+\frac{\sin x}{1}+\frac{\sin 2x}{2}+\ldots +\frac{\sin nx}{n}+\frac{\cos \left(n+\frac{1}{2}\right)x}{(2n+1)\sin \frac{x}{2}},

prove that for $0 < x < 2\pi$ the equality

\frac{\sin x}{1}+\frac{\sin 2x}{2}+\ldots +\frac{\sin nx}{n}+\ldots = \frac{\pi -x}{2}.

holds.

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Problem 1671

Applying Lagrange's formula to $\varphi (x)-\varphi \left(\dfrac {\pi }{2}\right)$ , where

\varphi (x) = \frac{\sin x}{1}+\frac{\sin 3x}{3}+\ldots +\frac{\sin (2n-1)x}{2n-1}+\frac{\cos 2nx}{4n\sin x},

prove that for $0 < x < \pi$ the equality

\frac{\sin x}{1}+\frac{\sin 3x}{3}+\frac{\sin 5x}{5}+\ldots +\frac{\sin (2n-1)x}{2n-1}+\ldots = \frac{\pi }{4}.

holds.

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Problem 1672

Using the binomial theorem, obtain the expansion

\frac{1}{\sqrt{1-x^{2}}} = 1+\frac{1}{2}x^{2}+\frac{1 \cdot 3}{2 \cdot 4}x^{4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}x^{6}+\ldots , \quad \left|x\right| < 1.

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Problem 1673

Prove that for $\left|x\right| < 1$ the equalities hold:

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

\frac{1}{(1-x)^{3}} = 1+3x+6x^{2}+10x^{3}+\ldots = \sum _{n=0}^{\infty } \frac{(n+1)(n+2)}{2}x^{n}.

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Problem 1674

Prove the equality:

\frac{1}{\sqrt{a^{2}+x^{2}}} = \frac{1}{a}-\frac{1}{2}\frac{x^{2}}{a^{3}}+\frac{1 \cdot 3}{2 \cdot 4}\frac{x^{4}}{a^{5}}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\frac{x^{6}}{a^{7}}+\ldots , \quad \left|x\right| < a.

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Problem 1675

Prove the equality:

(a+x)^{n} = a^{n}+nxa^{n-1}+\frac{n(n-1)}{1 \cdot 2}x^{2}a^{n-2}+\ldots , \quad \left|x\right| < a.

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Problem 1676

Expand the function $\sqrt{1+x}$ in powers of $x$ .

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Problem 1677

Expand the function $\ln \dfrac {a+x}{a-x}$ in powers of $x$ .

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Problem 1678

Expand the function $\ln (1-x+x^{2})$ in powers of $x$ .

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Problem 1679

Expand the function $\ln (2-3x+x^{2})$ in powers of $x$ .

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Problem 1680

Expand the function $\dfrac {(1+x)^{2}}{x}\ln (1+x)$ in powers of $x$ .

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Problem 1681

Prove the equality

\ln (1-2x\cos \varphi +x^{2}) = -2\sum _{n=1}^{\infty } \frac{\cos n\varphi }{n}x^{n}, \quad \left|x\right| < 1.

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Problem 1682

Verify that

\frac{1-a\cos x}{1-2a\cos x+a^{2}} = \sum _{n=0}^{\infty } a^{n}\cos nx, \quad \left|a\right| < 1.

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Problem 1683

Verify that

\frac{a\sin x}{1-2a\cos x+a^{2}} = \sum _{n=1}^{\infty } a^{n}\sin nx, \quad \left|a\right| < 1.

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Problem 1684

By a suitable change of variable, obtain the series expansion:

\ln \frac{x+1}{x-1} = 2\left(\frac{1}{x}+\frac{1}{3x^{3}}+\frac{1}{5x^{5}}+\ldots \right), \quad \left|x\right| > 1.

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Problem 1685

By a suitable change of variable, obtain the series expansion:

\ln x = 2\left(\frac{x-1}{x+1}+\frac{1}{3}\left(\frac{x-1}{x+1}\right)^{3}+\frac{1}{5}\left(\frac{x-1}{x+1}\right)^{5}+\ldots \right), \quad x > 0.

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Problem 1686

By a suitable change of variable, obtain the series expansion:

\frac{x}{\sqrt{1+x}} = \frac{x}{x+1}+\frac{1}{2}\left(\frac{x}{x+1}\right)^{2}+\frac{1 \cdot 3}{2 \cdot 4}\left(\frac{x}{x+1}\right)^{3}+\ldots , \quad x > -\frac{1}{2}.

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Problem 1687

Obtain the series expansion:

\ln \left(x+\sqrt{1+x^{2}}\right) = \sum _{n=0}^{\infty } (-1)^{n}\frac{1 \cdot 3 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot \ldots \cdot 2n}\frac{x^{2n+1}}{2n+1}, \quad \left|x\right| < 1.

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Problem 1688

Obtain the series expansion:

\ln \frac{2\left(1-\sqrt{1-x}\right)}{x} = \frac{1}{2} \cdot \frac{x}{2}+\frac{1 \cdot 3}{2 \cdot 4}\frac{x^{2}}{4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\frac{x^{3}}{6}+\frac{1 \cdot 3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8}\frac{x^{4}}{8}+\ldots , \quad \left|x\right| < 1.

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