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I want to use $\sum$ notation for this:$$\underbrace{\frac{1}{k}+\frac{1}{k}+\ldots +\frac{1}{k}+\frac{1}{k}}_{k\text{ times}}$$ I guessed$$\sum_1^k\frac{1}{k} ,$$but it equals $$1+\frac{1}{2}+\dots+\frac{1}{k-1}+\frac{1}{k}.$$

Silent
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    I often see $\sum^n{x}$ as a shorthand for $\sum_{i=1}^n{x}$, but in some contexts it could be ambiguos(e.g. in computer science indexes usually start at $0$, but you can disambiguate once at the beginning). – Bakuriu Oct 10 '13 at 12:00

3 Answers3

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You are getting confused because you are not being explicit about the index of the summation.

The summation $$\sum_{i=1}^k\frac{1}{k}$$ does represent $$\underbrace{\frac{1}{k}+\frac{1}{k}+\cdots+\frac{1}{k}}_{k\text{ times}}.$$ You're confusing this with the summation $$\sum_{i=1}^k\frac{1}{i}$$ which represents $$\frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{k}.$$

Zev Chonoles
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I think what you want is

$$\sum_{i=1}^k \frac{1}{k}.$$

Unless you want to multiply them in which case write

$$\prod_{i=1}^k \frac{1}{k}.$$

Why you want to do this, in any case, is beyond me.

JP McCarthy
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$$\sum_{i=1}^k\frac1i=1+\frac12+\cdots+\frac1{k-1}+\frac1k\quad\text{vs}\quad\sum_{i=1}^k\frac1k=\frac1k+\frac1k+\cdots+\frac1k+\frac1k=1$$

Did
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