I was solving a question in which I required to calculate $$\sum_{a=1}^{n} \frac{1}{a}$$
I asked to my teacher he told me that there is no formula to calculate it
So I want to know how to find $\sum_{a=1}^{n} \frac{1}{a}$
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What is the expression or quantity that is being summed? – SchrodingersCat Dec 23 '16 at 10:12
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http://math.stackexchange.com/questions/2234/sum-of-harmonic-progression There isn't a great closed form solution as such I guess. – Shraddheya Shendre Dec 23 '16 at 10:16
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To quote your teacher, "there is no formula [closed-form] to calculate it." What do you expect? – Clement C. Dec 23 '16 at 10:44
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Just add up the terms, that's how. :p – MPW Dec 23 '16 at 11:48
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i wanna sum for n=1000 or more than what – super saiyan Dec 23 '16 at 14:05
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Consider reading the comments above. – Clement C. Dec 23 '16 at 15:07
1 Answers
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If you just want to get close, you can approximate the sum by $\int_1^n \frac{1}{x} \; dx = \ln n.$ The error will be really close to the Euler constant $\gamma = .5772156649,$ so you can add that in too. E.g.,
$$\sum_{a=1}^{1000} \frac{1}{a} = 7.485470861.$$
and $\ln 1000 + \gamma = 7.484970943.$
There are refinements to the approximation, but we got 3 decimal places with just this.
