# The value of lim n → ∞ ∑ i = 1 n 2 n [ ( 2 i n ) 3 + 5 ( 2 i n ) ] .

### 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 45E
To determine

## To find: The value of limn→∞∑i=1n2n[(2in)3+5(2in)].

Expert Solution

The value of limni=1n2n[(2in)3+5(2in)] is 14.

### 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=1n2n[(2in)3+5(2in)] and obtain the limit as follows.

limni=1n2n[(2in)3+5(2in)]=limni=1n2n[8i3n3+10in]=limni=1n[16i3n4+20in2]=limn[16n4i=1ni3+20n2i=1ni]=limn[16n4(n(n+1)2)2+20n2(n(n+1)2)]

On further simplification the value of the limit becomes,

limni=1n2n[(2in)3+5(2in)]=limn[4(n+1)2n2+10n2n(n+1)]=limn[4n2(1+1n)2n2+10n2n2(1+1n)]=limn[4(1+1n)2+10(1+1n)]=4(1)+10(1)=14

Thus, value of limni=1n2n[(2in)3+5(2in)] is 14.

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