   Chapter 4.3, Problem 68E

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 →   ∞ 1 n ( 1 n + 2 n + 3 n + ... + n n )

To determine

To find:

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 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 sub interval,  a is the lower limit and b is the upper limit.

ii) Fundamental Theorem of Calculus: If f is continuous on [a, b], then

abf(x) dx=Fb-F(a)

Where, F is any anti derivative of f,

3) Given:

limn1n1n+2n+3n++nn

4) Calculation:

The given expression can be written as

limni=1n1n+2n+3n++nn·1n

limni=1nin·1n

Comparing with Riemann sum,

xi=i

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