I got this problem on my homework asking if the sequence a = $\dfrac{1}{n^2} + \dfrac{2}{n^2} + \dfrac{3}{n^2} + ... +\dfrac{n}{n^2}$, which I think is the same as asking what the sequence s = $\dfrac{1}{n}$ converges to. I am pretty confused as I thought the series of $\dfrac{1}{n}$ diverges? Could anyone help me with this problem?
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The series $\Sigma\dfrac1n$ diverges; what is $1+2+3+\cdots+n$? – J. W. Tanner Jan 09 '20 at 05:56
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Cf. this question and this question and this question – J. W. Tanner Jan 09 '20 at 06:04
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Note that the sum stops at $\frac n{n^2}$, which allows a finite value to be assigned to it.
For a fixed $n$ we can sum the terms and get $\frac{n(n+1)}{2n^2}=\frac12\cdot\frac{n+1}n$, whose limit as $n\to\infty$ is $\frac12$.
Parcly Taxel
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Note that there is a difference between series and sequence. These two words are confusing and interchangeable in nonmathematical language. But in mathematics, series is sequence obtained from a given sequence.
Series 1/n diverges but the sequence 1/n converges.
