Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/249

Substituting $x=1$ in (B), we get

$1-{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}-{\frac {1}{4^{2}}}+\cdots ,$

which is an alternating series that converges.

Substituting $x=-1$ in (B), we get

$1-{\frac {1}{2^{2}}}-{\frac {1}{3^{2}}}-{\frac {1}{4^{2}}}-\cdots ,$

which is convergent by comparison with the p series ($p>l$).

The series in the above example is said to have $[-1,1]$ as the interval of convergence. This may be written $-1\leq x\leq 1$, or indicated graphically as follows:

EXAMPLES

For what values of the variable are the following series convergent?

			Graphical representation of intervals of convergence[1]
15.	$1+x+x^{2}+x^{3}+\cdots .$	Ans. $-1<x<1.$
16.	$x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots .$	Ans. $-1<x\leq 1.$
17.	$x+x^{4}+x^{9}+x^{16}+\cdots .$	Ans. $0<x<1.$
18.	$x+{\frac {x^{2}}{\sqrt {2}}}+{\frac {x^{3}}{\sqrt {3}}}+\cdots .$	Ans. $-1\leq x<1.$
19.	$1+x+{\frac {x^{2}}{x!}}+{\frac {x^{3}}{3!}}+\cdots .$	Ans. All values of $x$.
20.	$1-{\frac {\theta ^{2}}{2!}}+{\frac {\theta ^{4}}{4!}}-{\frac {\theta ^{6}}{6!}}+\cdots .$	Ans. All values of $x$.
21.	$\phi -{\frac {\phi ^{3}}{3!}}+{\frac {\phi ^{5}}{5!}}-{\frac {\phi ^{7}}{7!}}+\cdots .$	Ans. All values of $\phi$.
22.	${\frac {\sin \alpha }{1^{2}}}-{\frac {\sin 3\alpha }{3^{2}}}+{\frac {\sin 5\alpha }{5^{2}}}+\cdots .$	Ans. All values of $\alpha$.

1. End points that are not included in the interval of convergence have circles drawn around them.