calculste A_n =∫_(−(1/2)) ^(1/2) x^n (√((1−x)/(1+x)))dx find nature of the serie Σ A


Question and Answers Forum

All Questions Topic List
Relation and Functions Questions
Previous in All Question Next in All Question
Previous in Relation and Functions Next in Relation and Functions
Question Number 98424 by mathmax by abdo last updated on 13/Jun/20

$calculste A_{n} = \int_{- \frac{1}{2}}^{\frac{1}{2}} x^{n} \sqrt{\frac{1 - x}{1 + x}} dx$ $find nature of the serie Σ A_{n}$
	Answered by maths mind last updated on 14/Jun/20
	$x=cos(2a) ⇒∫_((5π)/6) ^(π/6) cos^n (2a)(√(tg^2 (a))).−2sin(2a)da..A =∫−4cos^n (2a)tg(a)sin(a)cos(a)da =∫−4cos^n (2a)sin^2 (a) =2∫cos^n (2a)(cos(2a)−1)da =2∫_0 ^(π/6) {cos^(n+1) (2a)−cos^n (2a)}da =2∫cos^(n+1) (2a)−2∫cos^n (2a)da =∫cos^(n+1) (s)ds−∫cos^n (s)ds..I B(x;a,b)=∫_0 ^x t^(a−1) (1−t)^(b−1) dt ...incompletd Betta function ∫cos^n (t)dt, u=cos^2 (t) −2∫u^((n/2)+1) (1−u)^(1/2) du=−2β(u;(n/2)+2,(3/2))+c .−2β(cos^2 (t);(n/2)+2,(3/2))...I ∫_((5π)/6) ^(π/2) cos^n (t)dt..withe t=π−c ∫_(π/2) ^(π/6) (−1)^n cos^n (t) A=−2∫_((5π )/6) ^(π/2) cos^(n+1) (t)+2∫_((5π)/6) ^(π/2) cos^n (t)dt+2∫_(π/2) ^(π/6) cos^(n+1) (t)dt−2∫_(π/2) ^(π/6) cos^n (t)dt expresse withe..I ∣An∣≤∫∣x∣^n (√3)=(√3)(2^(n+1) /(n+1)) ⇒ΣA_n Cv absilutly ⇒ΣA_n Cv$
	$x = c o s (2 a)$ $\Rightarrow \int_{\frac{5 π}{6}}^{\frac{π}{6}} {c o s}^{n} (2 a) \sqrt{{t g}^{2} (a)} . - 2 s i n (2 a) d a . . A$ $= \int - 4 {c o s}^{n} (2 a) t g (a) s i n (a) c o s (a) d a$ $= \int - 4 {c o s}^{n} (2 a) {s i n}^{2} (a)$ $= 2 \int {c o s}^{n} (2 a) (c o s (2 a) - 1) d a$ $= 2 \int_{0}^{\frac{π}{6}} {{c o s}^{n + 1} (2 a) - {c o s}^{n} (2 a)} d a$ $= 2 \int {c o s}^{n + 1} (2 a) - 2 \int {c o s}^{n} (2 a) d a$ $= \int {c o s}^{n + 1} (s) d s - \int {c o s}^{n} (s) d s . . I$ $B (x; a, b) = \int_{0}^{x} t^{a - 1} {(1 - t)}^{b - 1} d t . . . i n c o m p l e t d B e t t a f u n c t i o n$ $\int {c o s}^{n} (t) d t, u = {c o s}^{2} (t)$ $- 2 \int u^{\frac{n}{2} + 1} {(1 - u)}^{\frac{1}{2}} d u = - 2 β (u; \frac{n}{2} + 2, \frac{3}{2}) + c$ $. - 2 β ({c o s}^{2} (t); \frac{n}{2} + 2, \frac{3}{2}) . . . I$ $\int_{\frac{5 π}{6}}^{\frac{π}{2}} {c o s}^{n} (t) d t . . w i t h e$ $t = π - c$ $\int_{\frac{π}{2}}^{\frac{π}{6}} {(- 1)}^{n} {c o s}^{n} (t)$ $A = - 2 \int_{\frac{5 π}{6}}^{\frac{π}{2}} {c o s}^{n + 1} (t) + 2 \int_{\frac{5 π}{6}}^{\frac{π}{2}} {c o s}^{n} (t) d t + 2 \int_{\frac{π}{2}}^{\frac{π}{6}} {c o s}^{n + 1} (t) d t - 2 \int_{\frac{π}{2}}^{\frac{π}{6}} {c o s}^{n} (t) d t$ $e x p r e s s e w i t h e . . I$ $∣ A n ∣⩽ \int ∣ x ∣^{n} \sqrt{3} = \sqrt{3} \frac{2^{n + 1}}{n + 1}$ $\Rightarrow Σ A_{n} C v a b s i l u t l y \Rightarrow Σ A_{n} C v$
	Commented by abdomathmax last updated on 15/Jun/20

	$thank you sir .$
	Commented by maths mind last updated on 15/Jun/20

	$w i t h e p l e a s u r$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum