# The value of lim n → ∞ ∑ i = 1 n 1 n ( i n ) 2 .

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter F, Problem 43E
To determine

## To find: The value of limn→∞∑i=1n1n(in)2.

Expert Solution

The value of limni=1n1n(in)2 is 13.

### Explanation of Solution

Theorem used:

Let c be a constant and n be a positive integer. Then,

i=1nc=nc, i=1ni=n(n+1)2 and i=1ni2=n(n+1)(2n+1)6.

Calculation:

Simplify the expression limni=1n1n(in)2 and obtain the limit as follows.

limni=1n1n(in)2=limni=1n1n(i2n2)=limni=1n1n3(i3)=limn1n3i=1ni3=limn1n3(n(n+1)(2n+1)6)

On further simplification the value of the limit becomes,

limni=1n1n(in)2=limn1n3(n2(1+1n)n(2+1n)6)=limn(1+1n)(2+1n)6=(1+0)(2+0)6[limn1n=0]=13

Thus, value of limni=1n1n(in)2 is 13.

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