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The function u(n) attains the values u...

The function `u_(n)` attains the values
`u_(1) = (1)/( 2) , u_(2) = ( 1)/( 2) +( 1)/( 2.4) + ( 1)/( 2.4.6)`
`u_(n) = ( 1)/( 2) + ( 1)/( 2.4) + "...." ( 1) /( 2.4 "...." ( 2n )) ,"............"`
Prove that `underset( n rarr oo) ( "lim") u_(n) lt 2`

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MOTION-LIMIT-EXERCISE-4
  1. The function u(n) attains the values u(1) = (1)/( 2) , u(2) = ( 1)/(...

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  2. Let alpha and beta be the distinct roots of ax^(2) + bx + c = 0 then u...

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  3. underset( x rarr 2 ) ( "lim") (( sqrt( 1- cos { 2 ( x - 2)}))/( x - 2...

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  4. underset( x rarr 0 ) ( "lim")( (1- cos 2x) ( 3 + cos x ))/( x tan 4x) ...

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  5. (lim)(xvec0)(sin(picos^2x))/(x^2) is equal to (1) pi/2 (2) 1 (3) ...

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  6. Let p = underset( x to 0^(+)) lim(1 + tan^(2) sqrt(x))^(1//2x) , th...

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  7. lim(x->pi/2)(cotx-cosx)/((pi-2x)^3)

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  8. For each t in R let [t] be the greatest integer less than or equal to ...

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  9. For x gt 0 , underset( x rarr 0) ( "Lim") [ ( sin x)^(1//x) + ((1)/( x...

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  10. Let L = lim(x rarr 0) ( a - sqrt( a^2 - x^2) - x^2/4)/x^4, a>0.If L is...

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  11. If underset( x to 0) lim [1+x log (1+b^(2))]^(1/x) = 2b sin^(2) the...

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  12. If underset( x rarr oo) ( "lim") (( x^(2) + x+ 1)/(x+1) -ax -b) = 4, t...

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  13. Let alpha(a) and beta(a) be the roots of the equation {(a+1)^(1//3)-1}...

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  14. The largest value of non-negative integer a for which underset( x rarr...

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  15. Let m and n be the two positive integers greater than 1 . If underset(...

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  16. Let alpha, beta in R such that lim(x ->0) (x^2sin(betax))/(alphax-sin...

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  17. Let f(x) = ((1 - x(1+ |1-x | )) /(|1-x|)) cos(1/(1-x)) for x!=1

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  18. Let f: R rarr R be a differentiable function with f(0)=0. If y=f(x) ...

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