use-weierstrass-m-test-and-dirichlet-test-to-confirm-the-uniformly-covergence-of-the-following-series-in-the-interval-0-1-a-n-1-cosnx-n-4-b-n-1-cosnx-n-8-7-c-n-1-

Question Number 133277 by Engr_Jidda last updated on 20/Feb/21

u s e w e i e r s t r a s s m - t e s t a n d d i r i c h l e t

t e s t t o c o n f i r m t h e u n i f o r m l y c o v e r g e n c e

o f t h e f o l l o w i n g s e r i e s i n t h e i n t e r v a l [0, 1]

a) \underset{n = 1}{\sum^{\infty}} \frac{c o s n x}{n^{4}}

b) \underset{n = 1}{\sum^{\infty}} \frac{c o s n x}{n^{\frac{8}{7}}}

c) \underset{n = 1}{\sum^{\infty}} \frac{x^{n}}{n^{\frac{3}{2}}}

d) \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{2} + x^{2}}

Commented by Dwaipayan Shikari last updated on 21/Feb/21

\underset{n = 1}{\sum^{\infty}} \frac{c o s n x}{n^{4}} = \frac{x^{3}}{48} (4 π - x) - \frac{π^{2}}{12} (x^{2} - \frac{2 π^{2}}{15})

\underset{n = 1}{\sum^{\infty}} \frac{1}{n^{2} + x^{2}} = \frac{π}{2 x} c o t h (π x) - \frac{1}{2 x^{2}}

Commented by Engr_Jidda last updated on 22/Feb/21

t h a n k y o u s i r