why do these 2 series have same interval of convergence

**tennisplaya999** · March 24th 2009, 02:36 PM

$\sum^{\infty}_{n=0} 1/(2+x)$

and
$\sum^{\infty}_{n=0} x^3/(x+2)$

oops they arent the series, but the power series representation so ignore the sigmas

**matheagle** · March 24th 2009, 09:20 PM

BECAUSE

${x^3\over x+2}= x^3 \biggl({1\over 2+x}\biggr) = x^3 \biggl({1/2\over 1-(-x/2)}\biggr)$

Which is a geo with r=-x/2, which converges whenever |-x/2|<1 or -2<x<2.

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