of n!x^n/(n^n) . i keep getting infinity as my answer and it says its wrong. what am i doing wrong?
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I am assuming you mean the series sum n = 1 to infinity of n!x^n/(n^n).
Use the ratio test to find the radius of convergence:
lim n -> infinity |[(n+1)!x^(n+1) / ((n+1)^(n+1))] / [n!x^n/(n^n)]|
= |x| lim n -> infinity [(n^n)(n+1) / ((n+1)^(n+1))]
= |x| lim n -> infinity (n/(n+1))^n
= |x| lim n -> infinity 1 / [((n+1)/n)^n]
= |x| lim n -> infinity 1 / [(1 + 1/n)^n]
= |x|(1/e), since lim n -> infinity (1 + 1/n)^n = e
The series converges if |x|/e < 1, or equivalently |x| < e.
The series diverges if |x|/e > 1, or equivalently |x| > e.
So the radius of convergence is R = e.
Lord bless you today!
Use the ratio test to find the radius of convergence:
lim n -> infinity |[(n+1)!x^(n+1) / ((n+1)^(n+1))] / [n!x^n/(n^n)]|
= |x| lim n -> infinity [(n^n)(n+1) / ((n+1)^(n+1))]
= |x| lim n -> infinity (n/(n+1))^n
= |x| lim n -> infinity 1 / [((n+1)/n)^n]
= |x| lim n -> infinity 1 / [(1 + 1/n)^n]
= |x|(1/e), since lim n -> infinity (1 + 1/n)^n = e
The series converges if |x|/e < 1, or equivalently |x| < e.
The series diverges if |x|/e > 1, or equivalently |x| > e.
So the radius of convergence is R = e.
Lord bless you today!