Convergence is an issue here!

Calculus Level 3

Find the radius of convergence R R for the series ∑ n = 1 ∞ ( 3 2 ) n n ( n + 1 ) 2 x n \displaystyle \sum_{n=1}^\infty \left(\dfrac32\right)^n \dfrac n{(n+1)^2} x^n .

Submit your answer as ⌊ 1000 R ⌋ \lfloor 1000R \rfloor .

Bonus : What happens to the series when ∣ x ∣ = R |x| = R and when x ∈ R x\in \mathbb R ?

Notation : ⌊ ⋅ ⌋ \lfloor \cdot \rfloor denotes the floor function .


The answer is 666.

Guillermo Templado by Guillermo Templado , 46, Spain

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1 solution

Otto Bretscher
May 14, 2016

The radius of convergence of ∑ a n x n \sum a_nx^n is R = lim ⁡ n → ∞ ∣ a n a n + 1 ∣ R= \lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right| if that limit exists. In our case this is the limit of ( 3 2 ) n n ( n + 1 ) 2 ( 2 3 ) n + 1 ( n + 2 ) 2 n + 1 , \left(\frac{3}{2}\right)^n\frac{n}{(n+1)^2}\left(\frac{2}{3}\right)^{n+1}\frac{(n+2)^2}{n+1}, which is 2 3 \frac{2}{3} . The answer is 666 \boxed{666} ... always a great answer, compañero ;)

At x = 2 3 x=\frac{2}{3} the series diverges by comparison with the harmonic series. At x = − 2 3 x=-\frac{2}{3} it converges to ln ⁡ ( 2 ) − π 2 12 \ln(2)-\frac{\pi^2}{12} ; write n ( n + 1 ) 2 = 1 n + 1 − 1 ( n + 1 ) 2 \frac{n}{(n+1)^2}=\frac{1}{n+1}-\frac{1}{(n+1)^2} .

I was afraid you would ask about x = 2 i 3 x=\frac{2i}{3} ;)

4 Helpful 0 Interesting 1 Brilliant 0 Confused

Yes,that is... and very well explained ↑ \uparrow , thank you very much, compañero!. I just only wanted to point that we can also use root criterion or Cauchy's criterion ( lim ⁡ n → ∞ ∣ a n ∣ n \displaystyle \lim_{n \to \infty} \sqrt[n]{|a_n|} ...) for calculating the radius of convergence R of this series, and at x = − 2 3 x = \frac{-2}{3} we can also use Dirichlet convergence criterion for seeing convergence at this point.

I like to talk to you because you have demonstrated me you are an authentic very good math teacher. This coming now are just two anecdotes that happened to me when I was a student (it is not necessary that you read it). Once, I heard a teacher of geometry and algebra saying that we are inhabitants of " The Plane". At that moment I laughed a lot, but two years later this man was one of the best teachers I've ever had, and two years later, I half understood what he meant... On another occasion, a teacher told us that mathematics was only to add, subtract, multiply, divide and finally and most importantly, know how to apply this 4 things. On the one hand, today, I understand what he meant , but on the other hand, I'm still not sure come to understand, because computers are better than we are calculating, and maybe in the not too distant future, even become better than humans applying them... Ok, I leave you, it was only for chatting a little bit, see you soon...

Guillermo Templado - 5 years, 1 month ago

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