How does one show S &#x223C;<!-- ∼ --> N 2 </msup> 4 </mfrac> (

Waldronjw

Waldronjw

Answered question

2022-07-12

How does one show S N 2 4 ( 2 ln ( N ) 1 ) ?

Answer & Explanation

Sanaa Hinton

Sanaa Hinton

Beginner2022-07-13Added 15 answers

If we study the series it's like to have
1 ( 1 ) + 2 ( 1 + 1 2 ) + + N ( 1 + 1 2 + 1 3 + + 1 N )
Which behaves like
n n H n = 1 4 ( 2 N 2 H N + 1 + 2 N H N + 1 N 2 N )
Asymptotically, we have
1 4 ( N 2 + 2 N 2 H N + 1 )
Now
H N + 1 ln ( N + 1 ) + γ
whence we can write
1 4 N 2 ( 1 + 2 ln ( N + 1 ) )

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