Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Test the series (n!)^2 / (n^n) for convergence.

diverges, the ratio test

Test the series 1/ n^(1+1/n) for convergence.

diverges, compare with the series 1/n

Let a[n] be a decreasing sequence of positive numbers.

(1) the series a[n] converges

(2) the series 2^k * a[2^k] converges

Prove that (2)=>(1).

(1) the series a[n] converges

(2) the series 2^k * a[2^k] converges

Prove that (2)=>(1).

page 44 in redcrest

page 58 in the first analysis notebook

page 58 in the first analysis notebook

Test the series 1/ 2^(sqrt(n)) for convergence.

converges, 2^k*a[2^k]

2^(A^k-k) > 2^k, where A>1

2^(A^k-k) > 2^k, where A>1

Test the series 1/ 2^log(n) for convergence.

diverges, 2^k*a(2^k)

Test the series 1/sqrt(n) * tan(1/sqrt(n)) for convergence.

diverges

hint: lim tan(x)/x = 1 (x->0). compare with 1/n

hint: lim tan(x)/x = 1 (x->0). compare with 1/n

Test the series ( 1/2 * arctan(n) )^n for convergence.

converges, Cauchy's test

Test the series (1-1/n)^(n^2) for convergence.

converges, Cauchy's test

b^x = y

Read in English: log(b,y) = x

Read in English: log(b,y) = x

x = the logarithm of y to the base b

Test the series 1/log(n!) for convergence.

diverges

show that the series 1/(n*log(n)) diverges,

and then use: 2^k*a(2^k)

show that the series 1/(n*log(n)) diverges,

and then use: 2^k*a(2^k)