anonymous
  • anonymous
I do not understand how to do this.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Write limn→∞∑k=1n(2+k∗(5/n))3∗5/n as a definite Integral
tkhunny
  • tkhunny
I can't understand the notation. \(\lim_{n\rightarrow \infty}ln(2+k(5/n))^{3}\cdot\dfrac{5}{n}\)?
anonymous
  • anonymous
yes

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anonymous
  • anonymous
that is the notation
tkhunny
  • tkhunny
Not quite. What is the "k=" doing in there?
anonymous
  • anonymous
im not sure thats what my problem says though. Think it stands for constant
tkhunny
  • tkhunny
'k' maybe, but '='?
anonymous
  • anonymous
oh im sorry let me rewrite equation real quick
tkhunny
  • tkhunny
Maybe you can use the [Draw] feature?
anonymous
  • anonymous
\[\lim_{n \rightarrow \infty}\sum_{k=1}^{n}(2+k*5/n)^3 *5/n\]
tkhunny
  • tkhunny
Well, that makes more sense!
tkhunny
  • tkhunny
Did you just learn, in that moment, just enough LaTeX to write that? Good work.
anonymous
  • anonymous
Thank you. Now i am unsure of how to convert that to a definite integral
tkhunny
  • tkhunny
Let's remember the definition of a Riemann Integral. Roughly, you chop things up into smaller and smaller pieces and never stop doing this. This is EXACTLY the function of 'n' in the given expression. \(n \rightarrow \infty\)
anonymous
  • anonymous
right
anonymous
  • anonymous
still to not understand how to convert
tkhunny
  • tkhunny
Try a couple values for n and see how it goes.
anonymous
  • anonymous
what to do you mean try n values?
tkhunny
  • tkhunny
Do n=1, then do n = 2. Look for anything familiar or consistent.
anonymous
  • anonymous
5*(7k)^3?
tkhunny
  • tkhunny
The first thing I noticed was that \(5/n\) is constant for a given value of n. This suggests an equivalent expression: \(\lim_{n\rightarrow\infty}\dfrac{5}{n}\sum_{k=1}^{n}\left[2+\dfrac{5k}{n}\right]^{3}\)
anonymous
  • anonymous
yes i agree with that
anonymous
  • anonymous
and the two is constant
anonymous
  • anonymous
can you just give me in converted? I like working back from the conversion
phi
  • phi
I think of the 5/n as an increment (e.g. dx ) the k* 5/n ranges from 5/n to 5 for very large n, 5/n is close to 0, so k*5/n represents a variable x going from 0 to 5
anonymous
  • anonymous
i do not understand what that means
phi
  • phi
It helps if you have the "big picture" of how integration is summation of lots of thin triangles.
phi
  • phi
*rectangles
tkhunny
  • tkhunny
\(=\lim_{x\rightarrow\infty}\dfrac{5}{n}\sum_{k=1}^{n}f\left(\dfrac{5k}{n}\right)\) \(=\lim_{x\rightarrow\infty}\dfrac{b-a}{n}\sum_{k=1}^{n}f(x^{*})\) \(=\int_{0}^{5}f(x)\;dx = \int_{0}^{5}\left(2+5x\right)^{3}\;dx\)
phi
  • phi
I think it's just (2+x)^3 inside the integral
anonymous
  • anonymous
okay! Thank you
tkhunny
  • tkhunny
Right. Typo. Thanks.

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