The series \(\sum_{n=1}^{\infty} \frac{(-1)^n \sin n x}{n^{\log _e n}},\) \(x ∈ \mathbb{R}\) converges

Question

Question

This question was previously asked in

CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)

Answer 1

Concept:

M_n test:

let a_n be series of real number then

\(\sum_{n=1}^{\infty} |an| \leq\)| M_n |

if M_nIs convergent then a_n is convergent

Calculation:

here

|a_n | = \(\sum_{n=1}^{\infty} \frac{ \sin n x}{n^{\log _e n}} \leq \) \(\frac{1}{n^{\log _e n}}\)

| a_n | \(\leq\) \(\frac{1}{n^2}\)

convergent for all x belong R .

Hence option (4) is correct

Download Solution PDF