What is the interval of convergence of $sum {(8 x)^n}/{n^{7}}$ ?

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Answer 1 · 2017-02-16T10:17:04

The series

$sum_(n=1)^oo (8x)^n/n^7$

is absolutely convergent for $x in [-1/8,1/8]$

We can use the ratio test to determine for which values of $x$ the series:

$sum_(n=1)^oo (8x)^n/n^7$

is convergent.

Evaluate:

$lim_(n->oo) abs( ( (8x)^(n+1)/(n+1)^7) / ((8x)^n/n^7) ) = lim_(n->oo) abs ( ((8x)^(n+1) ) / (8x)^n) (n/(n+1))^7 = 8 abs(x)$

so the series is absolutely convergent for $abs(x) < 1/8$ and divergent for $abs(x) > 1/8$ .

For $abs(x) = 1/8$ the test is indecisive and we need to analyze in detail:

$(1) x= 1/8$

$sum_(n=1)^oo 1/n^7$ is convergent based on the p-series test.

$(1) x= -1/8$

$sum_(n=1)^oo (-1)^n/n^7$ that is absolutely convergent based on the p-series test.

We can conclude that:

$sum_(n=1)^oo (8x)^n/n^7$

is absolutely convergent for $x in [-1/8,1/8]$

What is the interval of convergence of sum {(8 x)^n}/{n^{7}} sum {(8 x)^n}/{n^{7}} ?