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# Extended Mathmatical Study - Math Problem Example

## Summary

This paper provides a background research into geometric series. Further on it continues with notions of harmonic series and harmonic numbers in order to investigate the problem known as the Leaning Tower of Lire, or Book Stacking Problem set in the end of the paper.

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Mathematically geometric sequence can be written as follows:

Basic properties of the geometric sequence follow straight from (1) and (2) (Courant and Robbins, 1996, pp. 13-14). That is if common ratio is positive all the terms of the geometric progression will be positive. In case if the common ratio is negative the sequence will be alternating-sign. If the sequence will decay exponentially to zero. Under terms will increase to infinity (positive in case of r>1 and unsigned in case of r<1). Obviously in case of r=1 the sequence consists of the same terms, and under r=-1 terms even modulo will change sign.

The next important notion is a geometric series. From the perspective of geometric sequences it is a sum of terms of a geometric progression. Therefore we can write up an arbitrary geometric series as:

The next question that arises is the calculation of the sum of terms of geometric sequence, or the value of a geometric series. Straight calculus is efficient only if the series is finite and consists of several terms. In other cases it useful to multiply both sides of (3) by (1-r). As we can see the sum consists of the same terms with opposite signs except the first and the last one:

This is an expression of finding partial sums of series , known as Gabriel's staircase (Swain 1994, p. ...

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