Series and Operations on Them

Find the sum of the series $s = 1+2x+3x^{2}+4x^{3}+\ldots$, using the equality:

Find the sum of the series $s = 1+3x+5x^{2}+7x^{3}+\ldots$, by forming the products $xs$ and $x^{2}s$.

Noting that the coefficients of the series

are related by $a_{n+2}-2a_{n+1}+a_{n}=2$, where $a_{n}$ is the coefficient of $x^{n}$, find the sum of the series by multiplying it by $1-2x+x^{2}$.

In the series

each successive coefficient equals the sum of the three preceding ones. Find the sum of the series.

Multiply the two series: $s = 1+2x+3x^{2}+4x^{3}+\ldots$ and $s_{1} = 1-2x+3x^{2}-4x^{3}+\ldots$.

By direct multiplication, prove that

where $s_{m} = 1+mx+\dfrac {m(m-1)}{1 \cdot 2}x^{2}+\dfrac {m(m-1)(m-2)}{1 \cdot 2 \cdot 3}x^{3}+\ldots$.

By multiplying series, prove the equality

Prove the equalities:

By direct summation, find $S_{n}$ and verify that for $\left|x\right| < 1$

For Morgan's series

one can obtain (for $\left|x\right| \neq 1$)

and hence the series always converges, but

Prove this.

Find by direct differentiation the derivative of the series $x-\dfrac {x^{2}}{2}+\dfrac {x^{3}}{3}-\dfrac {x^{4}}{4}+\ldots$, $\left|x\right| < 1$.

Find by direct differentiation the derivative of the series $x-\dfrac {x^{3}}{3}+\dfrac {x^{5}}{5}-\dfrac {x^{7}}{7}+\ldots$, $\left|x\right| < 1$.

Find by direct differentiation the derivative of the series $x-\dfrac {x^{3}}{3!}+\dfrac {x^{5}}{5!}-\dfrac {x^{7}}{7!}+\ldots$.

Find by direct differentiation the derivative of the series $1+x+\dfrac {x^{2}}{1 \cdot 2}+\dfrac {x^{3}}{1 \cdot 2 \cdot 3}+\dfrac {x^{4}}{1 \cdot 2 \cdot 3 \cdot 4}+\ldots$.

Verify that

and from this obtain:

and

Prove that the sum of Gauss's hypergeometric series

satisfies the equation

where

The Bessel function $J_{0}(x)$ is defined by the series

Prove that it satisfies the differential equation

Prove that

satisfies the differential equation

Having formed the differential equation for the corresponding function, prove the series expansion for $\left|x\right| < 1$:

Having formed the differential equation for the corresponding function, prove the series expansion for $\left|x\right| < 1$:

Having formed the differential equation for the corresponding function, prove the series expansion for $\left|x\right| < 1$:

Prove the series expansion:

Prove the series expansion:

Prove that $\cos (\mu \arccos x) = A\sin \dfrac {\mu \pi }{2}+B\cos \dfrac {\mu \pi }{2}$, where $A$ and $B$ are the series of the two preceding problems (1710 and 1711).

Prove the series expansion:

From problem 1713 ($e^{m\arcsin x} = 1+mx+\sum_{\nu =1}^{\infty } \frac{m(m^{2}+1^{2})(m^{2}+3^{2})\ldots (m^{2}+(2\nu -1)^{2})}{(2\nu +1)!}x^{2\nu +1}+\sum_{\nu =0}^{\infty } \frac{m^{2}(m^{2}+2^{2})(m^{2}+4^{2})\ldots (m^{2}+4\nu^{2})}{(2\nu +2)!}x^{2\nu +2}$) obtain

By expanding $e^{e^{x}}$ in two ways, verify that

Noting that $\sec x$ is an even function, we may conclude that the expansion of $\sec x$ has the form

where $E_{n}$ are the corresponding coefficients (the Euler numbers). Multiplying term by term by the equality

obtain a recurrence formula for the Euler numbers:

Since $E_{0}=1$, it follows from this formula that

Verify this.

It is known that $1-\dfrac {x}{2}\ctg \dfrac {x}{2} = \sum \dfrac {B_{n}}{(2n)!}x^{2n}$, where $B_{n}$ are coefficients (called the Bernoulli numbers). Prove that for the Bernoulli numbers there exists the recurrence formula:

From this formula it follows that

Verify this.

Write the first five terms of the expansion in powers of $x$ for the function $\ln (1+e^{x})$.

Write the first five terms of the expansion in powers of $x$ for the function $e^{\sin x}$.

Write the first five terms of the expansion in powers of $x$ for the function $(1+x)^{x}$.

Write the first five terms of the expansion in powers of $x$ for the function $\dfrac {1}{e^{x}-1}-\dfrac {1}{x}$.

Prove that for small $\dfrac {b}{a^{2}}$ the approximate equality $\sqrt{a^{2}+b} \approx a+\dfrac {b}{2a}$ has an error approximately equal to $\dfrac {a}{8}\left(\dfrac {b}{a^{2}}\right)^{2}$, and use this equality to compute the values of the radicals:

Prove the approximate formula

and use it to compute the roots: $\sqrt[5]{245}$, $\sqrt[7]{129}$, $\sqrt[9]{515}$, $\sqrt[10]{1027}$.

There is the following approximate formula for extracting roots:

where $a^{n}$ is an exact power close to $N$. Prove that the error of this formula is approximately equal to $\dfrac {n^{2}-1}{12n^{3}}\left(\dfrac {b}{a^{n}}\right)^{3}$.

Using the formula of problem 1724 ($\sqrt[n]{N} = \sqrt[n]{a^{n}+b} \approx a+\dfrac {2ab}{2nN-(n+1)b}$), compute the following roots:

Using the binomial theorem and the equality

compute the value of $\sqrt[3]{2}$ to 10 digits.

Using the binomial theorem and the identity

compute $\sqrt{2}$ to 10 digits.

Using the series for $\ln (1+x)$ at $x = -\dfrac {1}{10}, -\dfrac {1}{25}$ and $\dfrac {1}{80}$, as well as the identities

compute the natural logarithms of the numbers $2, 3, 5$ and $10$ to 10 decimal places.

Prove that the quantities

have, respectively, the following series expansions:

Prove that the tabular difference of ordinary five-digit logarithms of numbers, i.e. $\lg (n+1)-\lg n$, for $1000 < n < 10000$ is approximately expressed by the formula $\Delta = 10^{5} \cdot \dfrac {M}{n} = \dfrac {43429}{n}$; $M = 0{,}43429$ is the modulus of common logarithms, $\Delta$ being expressed in hundred-thousandths.

Prove that the tabular difference of logarithms of sines given one minute apart, expressed in hundred-thousandths, can be given by the approximate formula

Prove that formulas of the same kind for sines and for logarithms of tangents have the form:

The approximate equalities $\sin x \approx x$, $\tg x \approx x$ have high accuracy for small angles (in some cases the more accurate formulas $\sin x \approx x-\dfrac {x^{3}}{6}$, $\tg x \approx x+\dfrac {x^{3}}{3}$ are needed). At what visual angle is the diameter of a factory chimney of radius 2 m seen from a distance of 10 km?

The disk of the Sun is seen at an angle of $30'$. How many times greater is the distance to the Sun than the diameter of the Sun?

A railway embankment has a grade of $0{,}012$. What angle does it make with the horizontal?

In a day the Earth traverses approximately an arc of $1^{\circ }$ of its orbit, whose radius is $150 \cdot 10^{6}$ km. Determine by how much the Earth's path deviates from a straight line in 1 second (27 km) and in 1 minute?

An arc of $1^{\circ }$ of the Earth's meridian is about 112 km. By how much is it longer than its chord?

Prove the approximate formula: $l = \sqrt{13h}$, where $h$ is the height of the observer above the horizon in meters, and $l$ is the distance to the horizon in kilometers.

In measuring the distance between two points, the reading on a stretched tape showed 12 m. Knowing that the sag of the tape was 20 cm, and taking the shape of the tape to be an arc of a circle, determine the distance between the points with a correction for the sag.

Using series, compute $\sin 1^{\circ }$ to within $10^{-5}$.

Using series, compute $\cos 1^{\circ }$ to within $10^{-5}$.

Using series, compute $\sin 10^{\circ }$ to within $10^{-4}$.

Compute $\pi$ to within $10^{-10}$ using the identity

Shanks computed $\pi$ to 707 digits using the identity of problem 1743 ($\dfrac {\pi }{4} = 4\arctg \dfrac {1}{5}-\arctg \dfrac {1}{239}$). How many terms in the expansions of $\arctg \dfrac {1}{5}$ and $\arctg \dfrac {1}{239}$ did he have to retain?

One of Huygens's rules for computing the circumference of a circle is equivalent to the equality

where $P_{n}$ and $p_{n}$ are the perimeters of the circumscribed and inscribed regular polygons, and $r$ is the radius of the circle. Prove that the error of this equality is approximately equal to $\dfrac {30}{n^{4}}$.