   Chapter 4.3, Problem 67E

Chapter
Section
Textbook Problem

# Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. lim n →   ∞ ∑ i = 1 n ( i 4 n 5 + i n 2 )

To determine

To evaluate:

The limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].

Explanation

1) Concept:

i) Riemann sum:

If is f is integrable on [a, b], then

abfxdx=limni=1nfxi x

where,

x= b-an and xi=a+i x

f(x) is called as integrand, n is the number of sub intervals,  a is the lower limit and b is the upper limit.

ii) If f is continuous on [a, b], then

abf(x) dx=Fb-F(a)

where, F is any anti derivative of f,

3) Given:

limni=1ni4n5+in2

4) Calculation:

Factor out 1n  from the given expression

limni=1ni4n4+in·1n

Rewriting,

limni=1n0+i4n4+in·1n

Hence,

a=0,x=1n

x= b-an

1n= b-0n

1= b

Therefore,   b=1

xi=a+i x

xi=0+i1n=in

Let x=in then

i4n4+in=<

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