Show that for every integer 1 + 1 4 + 1 9 + · ·...

Michelle Mendoza

Michelle Mendoza

Answered

2022-07-05

Show that for every integer 1 + 1 4 + 1 9 + · · · + 1 n 2 2 1 n

Answer & Explanation

Kiana Cantu

Kiana Cantu

Expert

2022-07-06Added 22 answers

Hint: Note that
1 2 2 + 1 3 2 + + 1 n 2 < 1 1 2 + 1 2 3 + + 1 ( n 1 ) n .
The expression on the right turns out to be a telescoping sum. For 1 ( i 1 ) i = 1 i 1 1 i
One could also do an induction version of the above proof, using the fact that 1 ( n + 1 ) 2 < 1 n 1 n + 1
ban1ka1u

ban1ka1u

Expert

2022-07-07Added 5 answers

Consider a Riemann sum approximation for
1 n 1 x 2 d x
by means of the partition { 1 , 2 , 3 , , n } and a right endpoint approximation. Since 1 x 2 is decreasing, the approximation will give an underestimate for the integral.
We obtain
1 4 + 1 9 + + 1 n 2 1 n 1 x 2 d x = [ 1 x ] 1 n = 1 1 n
The result in the problem follows immediately.

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