Maths-
General
Easy

Question

L t left parenthesis n rightwards arrow infinity right parenthesis left parenthesis 1 plus 2 to the power of 4 plus 3 to the power of 4 plus midline horizontal ellipsis plus n to the power of 4 right parenthesis divided by n to the power of 5 minus L t left parenthesis 1 plus 2 cubed plus 3 cubed plus.. plus n cubed right parenthesis divided by n to the power of 4 equals

  1. 1 divided by 5
  2. 1 divided by 30
  3. Zero
  4. 1 divided by 4

hintHint:

In this question we will use the formula of the sum of the series
1 to the power of 4 plus 2 to the power of 4 plus 3 to the power of 4 plus space.... space plus n to the power of 4 equals fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis left parenthesis 3 n squared plus 3 n minus 1 right parenthesis over denominator 30 end fraction
1 cubed plus 2 cubed plus 3 cubed plus.... n cubed equals space fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction to find the limit.

The correct answer is: 1 divided by 5


    In this question we have to find the value of limit as n rightwards arrow infinity of fraction numerator 1 to the power of 4 plus 2 to the power of 4 plus space... plus n to the power of 4 over denominator n to the power of 5 end fraction space minus space limit as n rightwards arrow infinity of fraction numerator 1 cubed plus 2 cubed plus 3 cubed... plus n cubed over denominator n to the power of 5 end fraction
    Step1: Using the formula of sum of numbers.
    1 to the power of 4 plus 2 to the power of 4 plus 3 to the power of 4 plus space.... space plus n to the power of 4 equals fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis left parenthesis 3 n squared plus 3 n minus 1 right parenthesis over denominator 30 end fraction
    1 cubed plus 2 cubed plus 3 cubed plus.... n cubed equals space fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction
    Step2: Putting the value of sum in the given expression.
    => limit as n rightwards arrow infinity of fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis left parenthesis 3 n squared plus 3 n minus 1 right parenthesis over denominator 30 n to the power of 5 end fraction minus space limit as n rightwards arrow infinity of fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 n to the power of 5 end fraction
    =>limit as n rightwards arrow infinity of fraction numerator left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis left parenthesis 3 n squared plus 3 n minus 1 right parenthesis over denominator 30 n to the power of 4 end fraction minus space limit as n rightwards arrow infinity of fraction numerator left parenthesis n plus 1 right parenthesis squared over denominator 4 n cubed end fraction
    =>limit as n rightwards arrow infinity of fraction numerator left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis left parenthesis 3 n squared plus 3 n minus 1 right parenthesis over denominator 30 n to the power of 4 end fraction minus space limit as n rightwards arrow infinity of fraction numerator n squared plus 2 n plus 1 over denominator 4 n cubed end fraction
    In first part of the expression the highest exponent of n in denominator is 4. and limit is n minus greater than infinity then any term in the numerator with exponent less than 4 will go to 0. While in second part of the expression the highest exponent in the denominator 3. So, any terms of n in numerator with exponent less than 3 will go to 0
    In first part of the expression the coefficient of highest exponent of n is 6.
    =>So, the limit will be 6 over 30 minus 0 equals 1 fifth

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