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Jaycee Mathis

Jaycee Mathis

Answered question

2022-05-20

Can the inequality i = 1 n 1 i < 2 n 1 be proved without induction?

Answer & Explanation

Leah Conley

Leah Conley

Beginner2022-05-21Added 12 answers

With telescoping, write
i = 1 n 1 i = 1 + i = 2 n 1 i .
Then use
2 ( i i 1 ) = 2 i + i 1 > 1 i
to conclude
i = 1 n 1 i < 1 + 2 i = 2 n ( i i 1 ) = 1 + 2 ( n 1 ) = 2 n 1.
tilfaen4a

tilfaen4a

Beginner2022-05-22Added 4 answers

Using Abel summation formula you have
k = 1 n 1 k = n n + 1 n t 2 t 3 / 2 d t
where [t] is the integer part of t. So, if n 2
k = 1 n 1 k < n + 1 n t 2 t 3 / 2 d t = 2 n 1.

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