Calculus of a Single Variable, Chapter 9, 9.8, Section 9.8, Problem 5

sum_(n=0)^oo (-1)^n(x^n)/(n+1)
To find the radius of convergence of a series sum a_n , apply the Ratio Test.
L = lim_(n->oo) |a_(n+1)/a_n|
L=lim_(n->oo) |((-1)^(n+1) x^(n+1)/((n+1)+1))/((-1)^n(x^n)/(n+1))|
L=lim_(n->oo) | (-1) * (x^(n+1)/(n+2))/(x^n/(n+1))|
L=lim_(n->oo) | (x^(n+1)/(n+2))/(x^n/(n+1))|
L= lim_(n->oo) |x^(n+1)/(n+2) * (n+1)/x^n|
L=lim_(n->oo) |(x(n+1))/(n+2)|
L = |x| lim_(n->oo) |(n+1)/(n+2)|
L=|x| * 1
L=|x|
Take note that in Ratio Test, the series converges when L <1.
Llt1
|x|lt1
By Ratio Test, the series converges when |x|<1.
Therefore, the radius of convergence of the given series is R=1 .

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