Question

Need someone good at determining Series' Convergence to check this.

Answer 1

\(\color{blue}{\sf Dirichlet's~Test:}\) (my notion of it)

Suppose I have the following series:
\(\color{#000000}{ \displaystyle \sum_{\rm n= a}^{\infty} A{\rm(n)} }\)

where \(\color{#000000}{ \displaystyle A{\rm(n)}=x{\rm(n)}\cdot z{\rm(n)} }\)

such that \(\color{#000000}{ \displaystyle x{\rm(n)} }\) is a real-number sequence, and \(\color{#000000}{ \displaystyle z{\rm(n)} }\) is a
complex-number sequence; is CONVERGENT IF,

(1) \(\color{#000000}{ \displaystyle x{\rm(n)} }\) is a monotonically decreasing sequence.
That is, \(\color{#000000}{ \displaystyle x{\rm(n)}>x{\rm(n+1)} ~~\forall\left\{\left.{\rm n}\right|~~{\rm n}\in[{\rm a},\infty),~~~{\rm n}\in\mathbb{Z}\right\} }\)


(2) There is a number M such that the absolute value of the
partial sum of \(\color{#000000}{ \displaystyle z{\rm(n)} }\) is bounded for any number of terms k.

That is,
\(\color{#000000}{ \displaystyle \left|\sum_{\rm n=a}^{k}z{\rm(n)}\right| \le M,~~\forall~{\rm k}\in\mathbb{ N}}\)

Answer 2

and when conditions (1) and (2) are satisfied, the series is then convergent with no exceptions ..... am I correct, or any even smallest errors anywhere?

Answer 3

@Michele_Laino when you come online, if you will, can you please post some feedback and corrections (if any) ?

Answer 4

You forgot \(x_n\to 0\).

Answer 5

(1)and(2) => the series converges.
no exception. https://en.wikipedia.org/wiki/Dirichlet's_test#Statement

Answer 6

\(\displaystyle \lim_{n\to \infty}{x(n)=0}\)

Answer 7

yes, I forgot that, thank you!