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$1(1+$ x$4)$

as the sum of a power series and find the interval of convergence.

Solution

We have the following equation.

$11$ x

$=1+$ x $+$ x$2+$ x$3+=$

n $=0$

xn $|$x$|<1$

Replacing x by

$$x$4$

in the equation above, we have the following.

$11+$ x$4$

$=$

$11($x$4)$

$=$

n$$

n $=0$

$=$

$(1)$nx$4$nn $=0$

$=1$ x$4+$

$$ x$12+$ x$16$

Because this is a geometric series, it converges when

$|$x$4|<,$

that is$,$

x$4<$

or

$|$x$|<.$

Therefore the interval of convergence is the open interval

$.($Of course, we could have determined the radius of convergence by applying the Ratio Test, but that much work is unnecessary here.$)$