Problems

Find the sums of the following series:

a) $\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\frac{1}{4\times 5}+\dots$;

b) $\frac{1}{1\times 2\times 3}+\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\dots$;

c) $\frac{0!}{r!}+\frac{1!}{(r-1)!}+\frac{2!}{(r-2)!}+\frac{3!}{(r-3)!}+\dots$ for $r\ge 2$.

Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of $d$. Prove that $d>30,000$.

Could it be that a) $\sigma (n)>3n$; b) $\sigma (n)>100n$?

Prove that for a real positive $\alpha$ and a positive integer $d$, $\lfloor \alpha /d\rfloor =\lfloor \lfloor \alpha \rfloor /d\rfloor$ is always satisfied.

Let the number $\alpha$ be given by the decimal:

a) $0.101001000100001000001\dots$;

b) $0.123456789101112131415\dots$.

Will this number be rational?

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.

Prove the irrationality of the following numbers:

a) $\sqrt{3}17$

b) $\sqrt{2}+\sqrt{3}$

c) $\sqrt{2}+\sqrt{3}+\sqrt{5}$

d) $\sqrt{3}3-\sqrt{2}$

e) $\cos {10}^{\circ }$

f) $\tan {10}^{\circ }$

g) $\sin 1^{\circ }$

h) ${\log }_{2}3$

Is it possible for

a) the sum of two rational numbers irrational?

b) the sum of two irrational numbers rational?

c) an irrational number with an irrational degree to be rational?

One of the roots of the equation $x^2+ax+b=0$ is $1+\sqrt{3}$. Find $a$ and $b$ if you know that they are rational.

Prove that the number $\sqrt{2}+\sqrt{3}+\sqrt{5}+\sqrt{7}+\sqrt{11}+\sqrt{13}+\sqrt{17}$ is irrational.