Page:Calculus Made Easy.pdf/210

 been told at what height above the origin the curve will begin, when $$x=0$$. So we write, as the equation of the curve drawn in Fig. 51,



Exercises XVI. (See page 262 for Answers.)

(1) Find the ultimate sum of $$\frac{2}{3} + \frac{1}{3} + \frac{1}{6} + \frac{1}{12} + \frac{1}{24} + $$ etc.

(2) Show that the series $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7}$$ etc., is convergent, and find its sum to $$8$$ terms.

(3) If $$\log_\epsilon(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + $$ etc, find $$\log_\epsilon 1.3$$.

(4) Following a reasoning similar to that explained in this chapter, find $$y$$,

(5) If $$\dfrac{dy}{dx} = 2x + 3$$, find $$y$$.