Extended Mathematical Study - Math Problem Example

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. Geometric sequence has been known in human history from the very ancient times. One of the first researchers to analyse geometric progressions and some of its properties was Euclid (Elements, books VIII and IX). A sequence is called geometric if every of its terms starting from the second can be found as product of the previous term and a number, different from 0 and same for the whole sequence, called common ratio. Mathematically geometric sequence can be written as follows: (1) , where r is the common ratio, and a is a scale factor depicting the initial value of a sequence. Additionally the recursive rule determining the geometric sequence can be written as: (2) (n is an integer). 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 rm), then we can generalize formula (6) and get: (7) . Now if we differentiate formula (6) by r we will receive an easy way of calculation of series : , on the one side and: ; on the other. Equating both sides and dividing by a we will get: (8) . Furthermore, multiplying both sides of (8) by r we will get: (9) . This is an expression of finding partial sums of series , known as Gabriel's staircase (Swain 1994, p. 209). While turning n to infinity in (9) we will acquire the formula for the whole sum of a series: Note that under |r|>1 the series diverges as we get an exponential function converging to infinity multiplied by linear function that also converges to infinity. However, when |r| Read More