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2 years ago

Write without the absolute value sign |x-3| if x>4

Mathematics

2 answers:

Arisa [49]2 years ago

8 0

Answer: 8

Explanation: if x = 8 or higher then x is greater than 4

Oxana [17]2 years ago

5 0

Answer: X= 8

Why: 8-3 = 5 and 5 is the next number of 4 and is therefore greater than 4

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Lim n→∞[(n + n² + n³ + .... nⁿ)/(1ⁿ + 2ⁿ + 3ⁿ +....nⁿ)]

Schach [20]

Step-by-step explanation:

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\displaystyle\lim_{n \to \infty} \frac{n + {n}^{2} + {n}^{3} + - - + {n}^{n} }{ {1}^{n} + {2}^{n} + {3}^{n} + - - + {n}^{n} }$

To, evaluate this limit, let we simplify numerator and denominator individually.

So, Consider Numerator

$\rm :\longmapsto\:n + {n}^{2} + {n}^{3} + - - - + {n}^{n}$

Clearly, if forms a Geometric progression with first term n and common ratio n respectively.

So, using Sum of n terms of GP, we get

$\rm \: = \: \dfrac{n( {n}^{n} - 1)}{n - 1}$

$\rm \: = \: \dfrac{ {n}^{n} - 1}{1 - \dfrac{1}{n} }$

Now, Consider Denominator, we have

$\rm :\longmapsto\: {1}^{n} + {2}^{n} + {3}^{n} + - - - + {n}^{n}$

can be rewritten as

$\rm :\longmapsto\: {1}^{n} + {2}^{n} + {3}^{n} + - - - + {(n - 1)}^{n} + {n}^{n}$

$\rm \: = \: {n}^{n}\bigg[1 +\bigg[{\dfrac{n - 1}{n}\bigg]}^{n} + \bigg[{\dfrac{n - 2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]$

$\rm \: = \: {n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]$

Now, Consider

$\rm :\longmapsto\:\displaystyle\lim_{n \to \infty} \frac{n + {n}^{2} + {n}^{3} + - - + {n}^{n} }{ {1}^{n} + {2}^{n} + {3}^{n} + - - + {n}^{n} }$

So, on substituting the values evaluated above, we get

$\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{\dfrac{ {n}^{n} - 1}{1 - \dfrac{1}{n} }}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}$

$\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{ {n}^{n} - 1}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}$

$\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{ {n}^{n}\bigg[1 - \dfrac{1}{ {n}^{n} } \bigg]}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}$

$\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{\bigg[1 - \dfrac{1}{ {n}^{n} } \bigg]}{\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}$

$\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{1}{\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}$

Now, we know that,

$\red{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to \infty} \bigg[1 + \dfrac{k}{x} \bigg]^{x} = {e}^{k}}}}$

So, using this, we get

$\rm \: = \: \dfrac{1}{1 + {e}^{ - 1} + {e}^{ - 2} + - - - - \infty }$

Now, in denominator, its an infinite GP series with common ratio 1/e ( < 1 ) and first term 1, so using sum to infinite GP series, we have

$\rm \: = \: \dfrac{1}{\dfrac{1}{1 - \dfrac{1}{e} } }$

$\rm \: = \: \dfrac{1}{\dfrac{1}{ \dfrac{e - 1}{e} } }$

$\rm \: = \: \dfrac{1}{\dfrac{e}{e - 1} }$

$\rm \: = \: \dfrac{e - 1}{e}$

$\rm \: = \: 1 - \dfrac{1}{e}$

Hence,

$\boxed{\tt{ \displaystyle\lim_{n \to \infty} \frac{n + {n}^{2} + {n}^{3} + - - + {n}^{n} }{ {1}^{n} + {2}^{n} + {3}^{n} + - - + {n}^{n} } = \frac{e - 1}{e} = 1 - \frac{1}{e}}}$

3 0

2 years ago

A box contains 12 1/2 square feet of tiles and costs $40.21. What is the price per square feet?

liraira [26]

The answer is $3.21.
To find the answer, divide the total cost by the number of sq ft to get the price per sq ft.

3 0

4 years ago

Read 2 more answers

david earns $1.50 per hour more than peter. together, they earn $940 if they both work 40 hours in a week. how much money per ho

Korolek [52]

Answer:

469.25

Step-by-step explanation:

peter=x

david=x+1.50

x+(x+1.50)=940

x=940-x-1.50

2x=940-1.50

2x=938.5

x=938.5/2

x=469.25

8 0

3 years ago

Which of the following expressions represents "three times the difference between t and y"?

ioda

Hi the awnser is 3(t-y) because you always do the math that is in the parentheses first!

6 0

3 years ago

A cylinder has a volume of 384 cubic inches and a height of 8 inches, what is the radius

Triss [41]

The answer is: 3.91 inches .
___________________________________________

Note: Volume of cylinder: V = (base area) * (height);

in which: V = volume = 384 in.³ ;
h = height = 8 in. ;
Base area = area of the base (that is; "circle") = π r² ;
in which; "r" = radius;
___________________________________________
Solve for "r" :
___________________________________________
V = π r² * (8 in.) ;

384 in.³ = (8 in.) * (π r²) ;
___________________________________________
Divide EACH SIDE of the equation by "8" ;
___________________________________________
(384 in.³) / 8 = [ (8 in.) * (π r²) in.] / 8 ;
___________________________________________
to get:
___________________________________________
48 in.³ = (π r²) in.² * in. ;
___________________________________________
↔ (π r²) in.² * in. = 48 in.³ ;
___________________________________________
Rewrite this equation; using "3.14" as an approximation for: π ;
__________________________________________________
(3.14 * r²) in.² * in. = 48 in.³
_______________________________________
Divide EACH SIDE of the equation by:

"[(3.14)*(in.²)*(in.)]" ; to isolate "r² " on one side of the equation;
(since we want to solve for "r") ;
_____________________________________________________
→ [(3.14 * r²) in.² * in.] / [(3.14)*(in.²)*(in.)] = 48 in.³ / [(3.14)*(in.²)*(in.)] ;
__________________________________________________
→ to get: r² = 48/3.14 ;
________________________
→ r² = 15.2866242038216561 ;
_______________________________________
To solve for "r" (the radius; take the "positive square root" of EACH side of the equation:
__________________________________________________
→ +√(r²) = +√(15.2866242038216561)
__________________________________________________
→ r = 3.9098112747064475286 ; round to 3.91 inches .
___________________________________________________

4 0

3 years ago

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