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## Hot answers tagged summation

### trying to solve the nested sum $\sum_{n=1}^{\infty} \frac{1}{1^2+2^2+\dots+n^2}$

We have \begin{align} \sum_{n=1}^\infty\left(\frac{1}{n}+\frac{1}{n+1}-\frac{4}{2n+1}\right) & = \sum_{n=1}^\infty\left(\frac{1}{n}-\frac{2}{2n+1}+\frac{1}{n+1}-\frac{2}{2n+1}\right)\\ &= \...
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### Evaluate $\lim_{n\to\infty}\sum_{k = n^3}^{(n+1)^3}\frac{1}{\sqrt[3]{k^2 +4k}}$

Define $$S(n):= \sum\limits_{k = n^3 }^{(n + 1)^3 } {f(k)}$$ I shall follow Gary's hint; but provide more details. Consider $f: \Bbb N \to \Bbb R$ given by $$f(k) = \frac{1}{(k^2 + 4k)^{1/3}}$$ Note ...
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\begin{align} \sum_{i=1}^n \sum_{j=i}^n 1 &\stackrel{1.}= \sum_{i=1}^n (n+1-i) \\ &\stackrel{3.}= (n+1)\sum_{i=1}^n 1 - \sum_{i=1}^n i \\ &\stackrel{1., 2.}= (n+1)n - n(n+1)/2 \\ &\... • 6,044 1 vote ### Double summation - How to solve for \sum_{j=i}^n 1? \begin{eqnarray} \sum_{j=i}^{n}1&=&\underbrace{1+1+\cdots+1}_{n-i+1\text{ times}}\\ &=&n-i+1\\ \sum_{i=1}^{n}\sum_{j=i}^{n}1&=&\sum_{i=1}^{n}n-i+1\\ &=&n(n+1)-\sum_{i=1}... 1 vote ### trying to solve the nested sum \sum_{n=1}^{\infty} \frac{1}{1^2+2^2+\dots+n^2}  Sorry, too long for a comment We can use integration, but we have to be careful while changing the order of operations. Dealing with diverging terms, we can easily get an additional finite ... • 6,284 1 vote ### Can you calculate 1+2+3+4+\ldots+k geometrically? Thanks all for pointing out the whole picture is not a triangle, I focused too much on the picture made of dots, neglecting the fact that the number '1' is a 'square', but not a dot, so combining them ... • 41 1 vote ### trying to solve the nested sum \sum_{n=1}^{\infty} \frac{1}{1^2+2^2+\dots+n^2}  Actually you do not need integrals. \sum_{k=1}^{2N+1}\frac{(-1)^{k+1}}{k}=\sum_{n=0}^{N}\frac{1}{2n+1}-\sum_{n=1}^{N}\frac{1}{2n}  converges to $\log(2)$ by the Maclaurin series of $\log(1+x)$ ...
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