Home > Calculus > Determining the Radius and Interval of Convergence for a Power Series

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

Answer 1

Paisley Johnson

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 in convergence.

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 therefore draw the following conclusion:

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

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

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Answer 2

Cameron Alexander

The interval of convergence for the series (\sum \frac{(8x)^n}{n^7}) is (-\frac{1}{8} < x \leq \frac{1}{8}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.