study tbe convergence of Σ_n ^∞ (1/(nln(1+n)))


Question and Answers Forum

All Questions Topic List
Relation and Functions Questions
Previous in All Question Next in All Question
Previous in Relation and Functions Next in Relation and Functions
Question Number 127778 by Bird last updated on 02/Jan/21

$s t u d y t b e c o n v e r g e n c e o f$ $\sum_{n}^{\infty} \frac{1}{n l n (1 + n)}$
	Answered by mnjuly1970 last updated on 02/Jan/21

	$\underset{n = 1}{\sum^{\infty}} \frac{1}{n l n (n + 1)} ⩾ \underset{n = 1}{\sum^{\infty}} \frac{1}{(n + 1) l n (n + 1)}$ $= \underset{n = 2}{\sum^{\infty}} \frac{1}{n l n (n)} \underset{t h e o r e m}{\overset{c a u c h y d e n s i t y}{\approx}} \underset{k = 1}{\sum^{\infty}} \frac{2^{k}}{2^{k} l n (2^{k})}$ $= \frac{1}{l o g (2)} \underset{k = 1}{\sum^{\infty}} \frac{1}{k} \to \infty$ $h a r m o n i c s e r i e s i s d i v e r g e n t . .$ $t h e n t h e o r g i n a l s e i e s i s d i v e r g e n t . . .$ $c o m p a r i s o n t e s t . . .$
	Answered by mindispower last updated on 02/Jan/21

	$f : x \to x l n (1 + x) p o s i t i v i n c r e a s e f u n c t i o n$ $\int_{1}^{\infty} f (x) d x a n d Σ \frac{1}{n l n (1 + n)} s a m n a t u r e$ $\int_{1}^{\infty} \frac{1}{x l n (1 + x)} d x = B y p a r t {[\frac{l n (x)}{l n (1 + x)}]}_{1}^{\infty} - \int_{1}^{\infty} \frac{l n (x)}{(1 + x) {l n}^{2} (1 + x)}$ $w h e n x \to \infty$ $\frac{l n (x)}{(1 + x) {l n}^{2} (1 + x)} \sim \frac{1}{(1 + x) l n (1 + x)}$ $x \to \frac{1}{(1 + x) l n (1 + x)} n o t c v i n + \infty$ $\int \frac{d x}{(1 + x) l n (1 + x)} = l n (l n (1 + x) \to \infty$ $\Rightarrow Σ \frac{1}{n l n (1 + n)} D v$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum