use weierstrass m−test and dirichlet test to confirm the uniformly covergence of the following series in the interval [0,1] a) Σ_(n=1) ^∞ ((cosnx)/n^4 ) b) Σ_(n=1) ^∞ ((cosnx)/n^(8/7) ) c) Σ_(n=1) ^∞ (x^n /n^(3/2) ) d) Σ


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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

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