n-1-m-log-n-n-3-2-

Question Number 63139 by Tawa1 last updated on 29/Jun/19

\underset{n = 1}{\sum^{m}} \frac{\log n}{n^{3 / 2}}

Commented by Tawa1 last updated on 30/Jun/19

please help with this sir

Test for convergence . \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{3} \sin^{2} n}

Commented by Tawa1 last updated on 30/Jun/19

God bless you sir

Commented by mathmax by abdo last updated on 30/Jun/19

i f y o u n e e d c o n v e r g e n c e l e t S_{m} = \sum_{n = 1}^{m} \frac{l n (n)}{n^{\frac{3}{2}}}

w e h a v e {l i m}_{n \to + \infty} S_{m} = \sum_{n = 2}^{\infty} \frac{l n (n)}{n^{\frac{3}{2}}} l e t φ (x) = \frac{l n (x)}{x^{\frac{3}{2}}} w i t h x > 1

w e h a v e φ^{^{'}} (x) = \frac{\frac{x^{\frac{3}{2}}}{x} - l n (x) \frac{3}{2} x^{\frac{1}{2}}}{x^{3}} = \frac{\sqrt{x} - \frac{3}{2} \sqrt{x} l n (x)}{x^{3}} = \frac{\sqrt{x}}{x^{3}} {1 - \frac{3}{2} l n (x)}

\Rightarrow \exists n_{0} > 0 / f o r x > n_{0} φ i s d e c r e a z i n g o n] x_{0}, + \infty [s o t h e s e r i e a n d

\int_{2}^{+ \infty} \frac{l n (x)}{x^{\frac{3}{2}}} d x h a v e t h e s a m e n a t u r e b u t

\int_{2}^{+ \infty} \frac{l n (x)}{x^{\frac{3}{2}}} d x =_{l n (x) = t} \int_{l n (2)}^{+ \infty} \frac{t}{e^{\frac{3}{2} t}} e^{t} d t = \int_{l n (2)}^{+ \infty} t e^{(1 - \frac{3}{2}) t} d t

= \int_{l n (2)}^{+ \infty} {t e}^{- \frac{t}{2}} d t t h i s i n t e g r a l c o n v e r g e s b e c a u s e {l i m}_{t \to + \infty} t^{2} (t e^{- \frac{t}{2}}) = 0 s o

\sum_{n = 1}^{\infty} \frac{l n (n)}{n^{\frac{3}{2}}} c o n v e r g e s .