Question
Download Solution PDFश्रेणी
\(\sum_{n=1}^{\infty} \frac{(-1)^n \sin n x}{n^{\log _e n}}, \quad x ∈ \mathbb{R}\)
अभिसरित होती है
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंप्रत्यय:
Mn परीक्षण:
मान लीजिए an वास्तविक संख्याओं की एक श्रेणी है, तब
\(\sum_{n=1}^{\infty} |an| \leq\)| Mn |
यदि Mn अभिसारी है, तो an अभिसारी है।
गणना:
यहाँ
|an | = \(\sum_{n=1}^{\infty} \frac{ \sin n x}{n^{\log _e n}} \leq \) \(\frac{1}{n^{\log _e n}}\)
| an | \(\leq\) \(\frac{1}{n^2}\)
सभी x ∈ R के लिए अभिसारी है।
इसलिए विकल्प (4) सही है।
Last updated on Jun 23, 2025
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