Smarandache Sequences, Stereograms and Series

Florentin Smarandache is truly a Renaissance man, having produced quality work in art, mathematics and literature. This is my fifth book where I expand on some of his mathematical ideas and there seems to be no end in sight. In chapter 1, several sequences created by concatenating all of the natural numbers are examined. By expressing the numbers in bases other than 10, a single type of concatenation can generate several different sequences. Upon examination, it can be seen that there are some significant differences between the sequences. Subsets of the natural numbers can be concatenated to create additional sequences and several are examined in chapter 2. The successive odd numbers and the successive even numbers are concatenated to form two of the sequences that are considered in detail. A stereogram is a two-dimensional image that contains a three-dimensional image. By looking at it long enough, the eyes lose focus in just the right way and the image appears. Simple Smarandache stereograms can be constructed using only characters and many different images are presented in chapter 3. The Smarandache functions can be used in combination with other mathematical operations to create many different infinite series. Many such series are examined in chapter four and conclusions whether they converge or diverge are reached.