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

What is the interval of convergence of #sum_1^oo ln(n)/e^n (x-e)^n #?

Answer 1

Elijah Abram

#abs((x-e)/e)<1#

#((x-e)/e)^n < log(n)((x-e)/e)^n <(n+1)((x-e)/e)^n=ed/(dx)((x-e)/e)^(n+1) #

Consequently, the series converges for

#abs((x-e)/e)<1#

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Answer 2

Isabella Ackerman

The interval of convergence for the series (\sum_{n=1}^\infty \frac{\ln(n)}{e^n}(x-e)^n) can be determined using the ratio test. By applying the ratio test, the interval of convergence can be found to be (|x - e| < e).

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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.