# The value of lim n → ∞ ∑ i = 1 n 1 n [ ( i n ) 3 + 1 ] .

### 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 44E
To determine

## To find: The value of limn→∞∑i=1n1n[(in)3+1].

Expert Solution

The value of limni=1n1n[(in)3+1] is 54.

### 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)3+1] and obtain the limit as follows.

limni=1n1n[(in)3+1]=limni=1n(i3n4+1n)=limn(i=1ni3n4+i=1n1n)=limn(1n4i=1ni3+1ni=1n1)=limn(1n4(n(n+1)2)2+1n(n))

On further simplification the value of the limit becomes,

limni=1n1n[(in)3+1]=limn(1n4(n4(1+1n)4)+1)=limn((1+1n)4+1)=14+1[limn1n=0]=54

Thus, value of limni=1n1n[(in)3+1] is 54.

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