For n a positive integer, set S ( n ) = 1 + 1/2 + 1/3 + ⋯ + 1/ ( 2 n − 1 ) . Which inequalities hold for every n?

Group the terms in blocks from 2 j to 2 j + 1 − 1 . Each block contributes at most 1 , giving S ( n ) ≤ n . In S ( 2 n ) , the initial term is 1 and each of the next 2 n − 1 blocks contributes more than 1/2 , so S ( 2 n ) > 1 + ( 2 n − 1 ) /2 > n .

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MathematicsSequence & SeriesSpecial sequences/seriesMedium2 minQB

MathematicsMediummultiple choice

For n a positive integer, set $S (n) = 1 + 1/2 + 1/3 + \dots + 1/ (2^{n} - 1)$ . Which inequalities hold for every n?

Options:(select one or more)

Answer:

A, D

Solution:

Group the terms in blocks from $2^{j}$ to $2^{j + 1} - 1$ . Each block contributes at most 1 , giving $S (n) \leq n$ . In $S (2 n)$ , the initial term is 1 and each of the next $2 n - 1$ blocks contributes more than $1/2$ , so $S (2 n) > 1 + (2 n - 1) /2 > n$ .

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:Special sequences/series

⏱ 2mℹ️ Source: QB

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