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]#
