# Questions tagged [summation-method]

Use for methods for constructing generalized sums of series, generalized limits of sequences, and values of improper integrals.

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### Double Abel summation

I am trying to calculate the Abel summation formula for a function of the form $$\sum _{j=1}^x \sum _{i=1}^x \phi \left( x-i j \right)$$ where the function $\phi$ meets the requirements for Abel's ...
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### Cesaro summation for matrices

I wonder if Cesaro summation for matrices is the same as summation for sequences. Cesaro summation for sequences means convergence of the arithmetic means (averages) of partial sums of sequence. For ...
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### Is there an easy way to sum $\sum_{1 \leq i < j < k \leq n} 1$?

I'd like to have a standard procedure to sum terms like $$\sum_{1\,\leq\, i\, <\, j\, <\, k\, \leq\, n} 1$$ without having to "telescope" the sum, beggining from the outermost one and ...
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### Fubini theorem in non classical summation methods

I am aware that in the theory of classical infinite sums , one can not generally interchange the order of a double sum or do other infinite sum manipulations. However, these infinite sum manipulations ...
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### How do I convert a function to sigma notation when there is a nested and lagging summation within the function?

I am trying to simplify the following function into summation form: $$f(x) = \frac{x_1}{x_0}+\frac{x_2}{x_0-x_1}+\frac{x_3}{x_0-x_1-x_2}+...+\frac{x_n}{x_0-x_1-x_2-...-x_{n-1}}.$$ However, I am unsure ...
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### Summation with products of values with different powers

For an exercise I am working on I have the following equation, $$c\sum_{n=1}^{\infty}n\sum_{j=1}^{n}(\frac{\lambda}{2\mu})^{j}(\frac{\lambda}{3\mu})^{n-j},$$ where $\lambda<\mu$. I have been ...
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