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

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_(n=1) ^∞ (1/(n^2 +x^2 ))

$${use}\:{weierstrass}\:{m}−{test}\:{and}\:{dirichlet} \\ $$$${test}\:{to}\:{confirm}\:{the}\:{uniformly}\:{covergence} \\ $$$${of}\:{the}\:{following}\:{series}\:{in}\:{the}\:{interval}\:\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.{a}\right)\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{cosnx}}{{n}^{\mathrm{4}} } \\ $$$$\left.{b}\right)\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{cosnx}}{{n}^{\frac{\mathrm{8}}{\mathrm{7}}} } \\ $$$$ \\ $$$$\left.{c}\right)\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{x}^{{n}} }{{n}^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$ \\ $$$$\left.{d}\right)\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} +{x}^{\mathrm{2}} } \\ $$$$ \\ $$

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

Σ_(n=1) ^∞ ((cosnx)/n^4 )=(x^3 /(48))(4π−x)−(π^2 /(12))(x^2 −((2π^2 )/(15)))  Σ_(n=1) ^∞ (1/(n^2 +x^2 ))=(π/(2x))coth(πx)−(1/(2x^2 ))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{cosnx}}{{n}^{\mathrm{4}} }=\frac{{x}^{\mathrm{3}} }{\mathrm{48}}\left(\mathrm{4}\pi−{x}\right)−\frac{\pi^{\mathrm{2}} }{\mathrm{12}}\left({x}^{\mathrm{2}} −\frac{\mathrm{2}\pi^{\mathrm{2}} }{\mathrm{15}}\right) \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} +{x}^{\mathrm{2}} }=\frac{\pi}{\mathrm{2}{x}}{coth}\left(\pi{x}\right)−\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} } \\ $$

Commented by Engr_Jidda last updated on 22/Feb/21

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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