What is the solution to this mathematical series of real numbers?

[Michael_0039]

Homework Statement

I used D'alembert criterion to determine if the mathematical sequence converges or diverges

Relevant Equations

Σ(1/(n+1)^2

 

[Delta2]

You have a mistake in the last line, it is easy to prove that ##\lim_{n \to +\infty}(1-\frac{1}{n+2})^{n+2}=e^{-1}=\lim_{n \to +\infty}(1-\frac{1}{n+2})^{n}## so the final limit is 1.

Instead upper bound the series by the hyperharmonic series (p-series for p=2 )which is a well know result that it converges, hence by the comparison criterion this series converges too.

 

[member 587159]

Comparing with the convergent series

$$\sum_{n=1}^\infty \frac{1}{n^2} =\frac{\pi^2}{6}$$ yields the result.

Alternatively, use the Cauchy condensation criterium or the integral test.

 

[Michael_0039]

Thanks for your answer. I wanted to use D'alembert criterion to see if a solution can be found.