sequence question

**manyarrows** · April 10th 2009, 07:11 PM

How do I interpret $\frac{1}{2^{(n-1)}+1}+\frac{1}{2^{(n-1)}+2}+\frac{1}{2^{(n-1)}+3}+...+\frac{1}{2^n}$

**Media_Man** · April 10th 2009, 07:29 PM

$2^{n-1}+k$ advances from $k=1$ up to $k=2^{n-1}$ , at which point $2^{n-1}+2^{n-1}=2^n$

**CaptainBlack** · April 11th 2009, 12:49 AM

Originally Posted by manyarrows

How do I interpret $\frac{1}{2^{(n-1)}+1}+\frac{1}{2^{(n-1)}+2}+\frac{1}{2^{(n-1)}+3}+...+\frac{1}{2^n}$

The sum of the recipricals of all the natural numbers greater than $2^{n-1}$ and less than or equal to $2^n$

CB

**stapel** · April 11th 2009, 04:25 AM

Originally Posted by manyarrows

How do I interpret $\frac{1}{2^{(n-1)}+1}+\frac{1}{2^{(n-1)}+2}+\frac{1}{2^{(n-1)}+3}+...+\frac{1}{2^n}$

What do you mean by "interpret"? Were the instructions something like "describe the rule for the n-th term in words"...?

Thank you!

**manyarrows** · April 11th 2009, 11:45 PM

No it was to verify the inequality with that series > or = to 1/2.
How do I copy an equation from a previous thread and edit it in a new thread? Anyway , I have problem with sequences or series when the last term differs from the first. I don't know they look. For example in the above how does n=3 look compared to n=5? How do you incorporate the last term.

**NonCommAlg** · April 12th 2009, 12:27 AM

$\sum_{k=1}^{2^{n-1}} \frac{1}{2^{n-1} + k} \geq \sum_{k=1}^{2^{n-1}} \frac{1}{2^{n-1} + 2^{n-1}}=\frac{1}{2^n} \sum_{k=1}^{2^{n-1}} 1 = \frac{2^{n-1}}{2^n}=\frac{1}{2}.$

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