What is the following limit?

lim as n goes to infinity of (pi/n) (sin(pi/n) + sin(2pi/n) + sin(3pi/n) +...+ sin(npi/n)) =
I.) lim as n goes to infinity sigma (n and k=1) of pi/n sin(kpi/n)
II.) Definite integral from 0 to pi of sin(x)dx
III.) 2

A.) I only
B.) II only
C.) III only
D.) II and III only
E.) I, II, and III

Wow, I am really lost on this one, please help!

It's I, II, and III

Okay, I still haven't figured it out, but figured that option II is equal to 2. So, if II is true, then III must be true too, so this narrows it down to A, D, and E, if this helps any.

To determine the limit as n approaches infinity of the given expression, let's break it down step-by-step:

Step 1: Rewrite the given expression as a sum using sigma notation.
Starting with the expression, we can rewrite it as a sum using sigma notation:
(pi/n)(sin(pi/n) + sin(2pi/n) + sin(3pi/n) + ... + sin(npi/n)) = (pi/n) * sigma (from k=1 to n) of sin(kpi/n).

Step 2: Simplify the inside expression.
Next, let's simplify the expression inside the sum by dividing the argument of sin by n:
sin(kpi/n) = sin((k/n)pi).

Step 3: Substitute the simplified expression into the sum.
Now, substitute the simplified expression into the sum:
(pi/n) * sigma (from k=1 to n) of sin((k/n)pi) = (pi/n) * sigma (from k=1 to n) of sin(kpi/n).

Step 4: Relate the sum to a definite integral.
We can relate the sum to a definite integral by recognizing that as n approaches infinity, the sum becomes a Riemann sum. The Riemann sum is an approximation of an integral.
Therefore, the expression can be rewritten as:
Lim as n approaches infinity of (pi/n) * sigma (from k=1 to n) of sin(kpi/n) = integral (from 0 to pi) of sin(x) dx.

Step 5: Evaluate the definite integral.
Now, we need to evaluate the definite integral of sin(x) from 0 to pi. The integral of sin(x) is -cos(x), so evaluating the definite integral, we get:
integral (from 0 to pi) of sin(x) dx = -cos(x) (evaluated from 0 to pi) = -cos(pi) - (-cos(0)) = -(-1) - (-1) = 2.

Step 6: Determine the limit.
Finally, we have determined that the limit as n approaches infinity of the given expression is equal to the value of the definite integral, which is 2.

Therefore, the answer is C.) III only, as only the third option correctly represents the limit.

To find the limit of the given expression, we can break it down into separate parts and analyze each one.

I.) We have a sigma notation in the expression, which represents a sum. The expression σ(n and k=1) of pi/n sin(kpi/n) represents the sum of pi/n sin(kpi/n) as k ranges from 1 to n. This is the same as the sum of the terms inside the parentheses in the original expression. Therefore, the limit as n goes to infinity of this sum is equivalent to the limit of the original expression.

II.) The definite integral from 0 to pi of sin(x)dx represents the area under the curve of the function sin(x) from 0 to pi. This integral represents a finite value of 2. Therefore, the limit of this definite integral as n goes to infinity is also 2.

III.) The value of 2 is a constant, independent of n. Therefore, the limit as n goes to infinity of 2 is also 2.

Putting it all together, we see that options II and III are both true. Therefore, the correct answer is D.) II and III only.