been told at what height above the origin the curve will begin, when ${\displaystyle x=0}$. 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 ${\displaystyle {\frac {2}{3}}+{\frac {1}{3}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{24}}+}$ etc.

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

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

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

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