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

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

Answer 1

Emily Ammerman

The series is not convergent for any value of #x#.

Apply the ratio test and assess:

#abs(a_(n+1)/a_n) = abs ( (( (n+1)! (8x-7)^(n+1))/7^(n+1))/(( n! (8x-7)^n)/7^n))#

#abs(a_(n+1)/a_n) = abs ( ((n+1)!)/(n!) 7^n/7^(n+1) (8x-7)^(n+1)/(8x-7)^n)#

#abs(a_(n+1)/a_n) = (n+1)/7 abs (8x-7)#

For any value of #x# we have therefore:

#lim_(n->oo) abs(a_(n+1)/a_n) = oo#

in addition to the series not convergent.

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Answer 2

Emma Aldridge

The interval of convergence for the given series is (-∞, ∞).

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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.