Question

@loveblue

Answer 1

@lovekblue the continuation is here. i'll write the solution again.

Answer 2

\[\Large \frac{ 2 }{ n }\left[ \left( \frac{ 3i }{ n } \right)^3+\frac{ 5i }{ n } +2 \right]=\frac{ 1 }{ n }\left[ 2\left( \frac{ 3i }{ n } \right)^3+\frac{ 10i }{ n } +2 \right]\]

Answer 3

\[=\frac{ 1 }{ n }\left[ 2\cdot3^3\left( \frac{ i }{ n } \right)^3+10\frac{ i }{ n } +4 \right]\]\[=\frac{ 1 }{ n }\left[ 54\left( 0+i\frac{ 1 }{ n } \right)^3+10(0+i\frac{ 1 }{ n }) +4 \right]\]

Answer 4

it's easier to deal with 1/n than 2/n. i'm "cheating" my way to solving it. LOL

Answer 5

set \[\large c_i=0+i \frac{1}{n}\] we conclude that \(a = 0\)

\[\Delta x = \frac{b-0}{n} = \frac{1}{n}\] we see that \(b = 1\)

finally, the expression inside the pair [ ] (which is \(f(c_i)\) is
\[54(c_i)^3+10c_i+4=f(c_i)\]
our \(f(x)\) is \[\Large f(x)=54x^3+10x+4\] and \[\Large \lim_{n \rightarrow \infty}\sum_{i=1}^{n}\frac{ 2 }{ n }\left[ \text{whatever is here}\right]=\int\limits_{0}^{1}\left( 54x^3+10x+4 \right) dx\]

Answer 6

@_@ oh wowww

Answer 7

can u give me a problem to let me try this out ~

Answer 8

while the solution is impressive, you will find that direct evaluation of the limit is far easier and less analytic.
let me show how.
(3i/n)^3=(27/n^3) i^3
(5i/n) = (5/n) i

now, summation (i) = n(n+1)/2
summation (i^3) = n^2(n+1)^2/4
summation (2) = 2n

Answer 9

thus, (2/n) [ (27/n^3) n^2(n+1)2/4 + (5/n) n(n+1)/2 + 2n]
\[=\frac{ 27n^2(n+1)^2 }{ 4n^4 }+\frac{ 5n(n+1) }{ 2n^2 }+\frac{ 2n }{ n }\]

evaluating the limit of each fraction at inf,
\[=\frac{ 27 }{ 4 }+\frac{ 5 }{ 2 }+2=\frac{ 45 }{ 4 }\]

Answer 10

but will they always have the same power of coefficient

Answer 11

it only works if they have the same power

Answer 12

it will always be the case for the riemann sum over a polynomial f(x)

Answer 13

ALWAYS!

Answer 14

A L W A Y S ! ! !

Answer 15

really, okay ;D

Answer 16

u basically covered 90% of the topics i've learned in this half year :P

Answer 17

some mistake there...
i forgot the 2 in (2/n). the correct answer should be 45/2.

Answer 18

what...oh lol..opps..me too >.>