Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/250

23.	${\frac {\cos x}{e^{x}}}+{\frac {\cos 2x}{e^{2x}}}+{\frac {\cos 3x}{e^{3x}}}+\cdots .$	Ans. $x>0.$.
Hint. Neither the sine nor cosine can exceed 1 numerically.
24.	$1+x\log a+{\frac {x^{2}\log ^{2}a}{2!}}+{\frac {x^{3}\log ^{3}a}{2!}}+\cdots .$	Ans. All values of $x$.
25.	${\frac {1}{1+x}}+{\frac {1}{1+x^{2}}}+{\frac {1}{1+x^{3}}}+\cdots .$	Ans. $x>1.$
26.	$x+{\frac {1}{2}}\cdot {\frac {x^{3}}{3}}+{\frac {1}{2}}\cdot {\frac {3}{4}}\cdot {\frac {x^{5}}{5}}+{\frac {1}{2}}\cdot {\frac {3}{4}}\cdot {\frac {5}{6}}\cdot {\frac {x^{7}}{7}}+\cdots .$	Ans. $-1\leq x\leq 1.$
27.	$1+x+2x^{2}+3x^{3}+\cdots .$
28.	$x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots .$
29.	$10x+100x^{2}+1000x^{3}+\cdots .$
30.	$1+x+(2!)x^{2}+(3!)x^{3}+\cdots .$

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