let-f-n-t-t-n-1-sin-n-with-t-from-0-1-and-from-0-pi-1-prove-the-uniform-convergence-of-f-n-t-on-0-1-2-let-S-t-f-n-t-calculate-0-1-S-t-dt-

Tinku Tara June 4, 2023 Relation and Functions 0 Comments

Question Number 48009 by maxmathsup by imad last updated on 18/Nov/18

let f_n (t)=t^(n−1) sin(nθ) with t from[0,1[ and θ from [0,π[ 1) prove the uniform convergence of Σ f_n (t) on [0,1[ 2) let S(t)=Σ f_n (t) calculate ∫_0 ^1 S(t)dt.

$${let}\:\:\:{f}_{{n}} \left({t}\right)={t}^{{n}−\mathrm{1}} {sin}\left({n}\theta\right)\:{with}\:{t}\:{from}\left[\mathrm{0},\mathrm{1}\left[\:{and}\:\:\theta\:{from}\:\left[\mathrm{0},\pi\left[\right.\right.\right.\right. \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{uniform}\:{convergence}\:{of}\:\Sigma\:{f}_{{n}} \left({t}\right)\:{on}\:\left[\mathrm{0},\mathrm{1}\left[\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{S}\left({t}\right)=\Sigma\:{f}_{{n}} \left({t}\right)\:\:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {S}\left({t}\right){dt}. \\ $$

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