25. Let f (x) be a continuous function. Express lim n ? ? 1 n [ f ( 1 n ) + f (

Question

25. Let f (x) be a continuous function. Express lim n ? ? 1 n [ f ( 1 n ) + f ( 2 n ) + ? + f ( n n ) ] as a definite integral. here is exercise 25, please be clear to receive all points

26. Use the result of Exercise 25 to evaluate

a. lim n ? ? 1 n 2 ( 2 + 4 + 6 + ? + 2 n ) ,

b. lim n ? ? 1 n 16 ( 1 15 + 2 15 + 3 15 + ? + n 15 ) ,

c. lim n ? ? 1 n ( sin ? n + sin 2 ? n + sin 3 ? n + ? + sin n ? n ) .

What can be said about the following limits?

d. lim n ? ? 1 n 17 ( 1 15 + 2 15 + 3 15 + ? + n 15 )

e. lim n ? ? 1 n 15 ( 1 15 + 2 15 + 3 15 + ...+ n 15 )

Please be clear to recieve full credit

Explanation / Answer

25)

lim n ? ? 1/n [ f ( 1/n ) + f ( 2/n ) + ? + f ( n/n ) ]

dx = 1/n

a = 0

b = 1

So the limit is equal to: int f(x) dx (x from 0 to 1)

26)

a) lim n ? ? 1/n^2 ( 2 + 4 + 6 + ? + 2 n ) =

lim n ? ? 1/n ( 2/n + 4/n + 6/n + ? + 2 n/n )

-> using result of problem 25 : f(x) = 2x

-> lim = int 2x dx = x^2 (0<x<1) = 1^2 - 0^2 = 1

-> lim n ? ? 1/n^2 ( 2 + 4 + 6 + ? + 2 n ) = 1

b)

lim n ? ? 1/n^16 ( 1^15 + 2^15 + 3^15 + ? + n^15 ) =

lim n ? ? 1/n ( (1/n)^15 + (2/n)^15 + (3/n)^15 + ? + (n/n)^15 ) =

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