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let m and k be positive integers (m not equal to k) Then lim n ---> infinty n { (1+1/n) m + ( 1 + 2/n) m + …..+ (1+k/n) m – k} is:

let m and k be positive integers (m not equal to k)
Then
limn ---> infinty n { (1+1/n)+ ( 1 + 2/n)m + …..+ (1+k/n)m – k} is: 
 

Grade:11

1 Answers

Aditya Gupta
2075 Points
one year ago
let y = 1/n so y tends to 0+
so L= lim y tends to 0+ { (1+y)+ ( 1 + 2y)m + …..+ (1+ky)m – k}/y
now write – k= – 1 – 1 …...... – 1 (k times)
L= lim y tends to 0+ [{(1+y)– 1} + {( 1 + 2y)m – 1}+ …..+ {(1+ky)m – 1}]/y
= lim y tends to 0+ {(1+y)– 1}/y + lim y tends to 0+ {(1+2y)– 1}/y + …...... + lim y tends to 0+ {(1+ky)– 1}/y
now, note that lim y tends to 0+ {(1+py)– 1}/y= mp (by l hopital)
so, L = 1*m + 2*m + …..... + k*m
mk(k+1)/2
kindly approve :)

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