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What's the value of- 1) 1 + 58 + 5.88.12 + 5.8.118.12.16+..........................∞ 2)12 + 1.32.4(14) + 1.3.52.4.6(14)2 + ............∞
post the solution if possible
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3 Answers
23(1-x)^{-n}=1+nx +\frac{n(n+1)}{1.2}x^{2}+\frac{n(n+1)(n+2)}{1.2.3}x^{3}+...
put n=53 , x = 3y4
(1-\frac{3y}{4})^{-\frac{5}{3}}=1+\frac{5}{4}y+\frac{5.8}{1.2}y^{2}+\frac{5.8.11}{4.8.12}y^{3}+...
hence
\int(1-\frac{3y}{4})^{-\frac{5}{3}}dy=y+\frac{5}{4.2}y^{2}+\frac{5.8}{4.8.3}y^{3}+\frac{5.8.11}{4.8.12.4}y^{4}+...
i.e
\int(1-\frac{3y}{4})^{-\frac{5}{3}}dy=y+\frac{5}{8}y^{2}+\frac{5.8}{8.12}y^{3}+\frac{5.8.11}{8.12.16}y^{4}+...
now put y = 1
hence ans is \int(1-\frac{3y}{4})^{-\frac{5}{3}}dy
at y = 1 wich u can do easily
for 2nd think something similar ( i think it involves differentiation )