Integration by Parts and Infinite Series

In this paper, by using a student’s calculation error in applying the integration formula using the subtraction method and reaching the correct answer, we reached the following formula ∫ fg = Σ1 k=0(-1)kf(k)g-(k+1) where g-(k) is the k th successive antiderivative of g and f(k) is the k th derivative of ffi. We find by using the above series and Maclaurin series, interesting results about the convergence of some series. Also, due to the introductory nature of the integration formula using the subtraction and the infinite series, this series can be new topics for the next research is to provide researchers with a new method for deriving series and discovering new series.