Thesis (Bachelor of Science in Mathematics) -- University of Rizal System-Morong.

EXECUTIVE SUMMARY This study entitled Application of Alternative Solution in Solving Taylor and Maclaurin Series was conducted at the University of Rizal System, Morong, Rizal during the academic year 2015-2016. The purpose of this study was to present and introduce the Taylor and Maclaurin series, and apply the alternative solution in solving finite series. Taylor series is a representation of a function as an infinite of terms that are calculated from the values of function's derivative at a single point, and a Maclaurin series is a taylor series expansion of a function centered at 0. The Alternative solution in solving finite series has been applied in two types of series by cases, Taylor series as case 1 and Maclaurin series as case 2. The researchers computed by Taylor series of three functions, exponential function, sin x and cos x, from 3 terms sequence and limited up to 20 terms sequence of the series. True to all values of X, the existing method and the alternative solution has an equal result on the 3 terms sequence of the series, regardless if the terms are negative or non-negative. As the number of terms increases the results obtained using the existing method repeats itself several times. It is because each term on the series has an n! (n factorial) as its divisor, and as the value of the factorial increases as the number of derivation occurs, the terms tend to be below 0 which can be approximately 0 and that makes the result to remain as it is. Moreover, on the alternative solution, the results show no clear pattern as how it is, it appears that it may be increasing, decreasing or alternating but not equal after the 3 terms sequence. Maclaurin series. Three functions have been computed for the Maclaurin series, sin x, exponential function and logarithmic function, from 3 terms sequence. True to all values of x, the existing method and the alternative solution n solving finite series obtained an equal result on non-negative. The results on the existing method have the same manner as the results in the Taylor series, the n! has something to do as why the results have been repeatedly a it is. Furthermore, the results obtained using the alternative solution is the same as in the Taylor series, it has no clear pattern and may sometimes be increasing, decreasing or alternating with relation to the number of terms. The researchers formulated a generalization which states that for instances, the alternative solution gives equal result with the existing method. when the given series has three terms, in cases that involved Taylor series, the given series is equal regardless if it's positive or negative values of x; and on the cases that involves Maclaurin series, the series should be non-negative. The results obtained on the n-terms higher than 3 terms, using the existing method and the alternative solution is not equal, both cases of Taylor and Maclaurin series