Question

How to prove $\sum_{n\ge 0} a_n x^n = \sum_{n\ge 0} \left(\sum_{k=0}^{n}(-1)^{k+1} \binom{n}{k}a_k\right) \frac{x^n}{(x-1)^{n+1}}$?

score 62 · Answer 1 · 2023-06-18 01:56:47

62

Views

How to prove $\sum_{n\ge 0} a_n x^n = \sum_{n\ge 0} \left(\sum_{k=0}^{n}(-1)^{k+1} \binom{n}{k}a_k\right) \frac{x^n}{(x-1)^{n+1}}$?

Published on 18 Jun 2023 - 1:56

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50

Views

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80

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67

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11

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103

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score 27 · Answer 15 · 2023-07-18 18:43:30

27

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List Question

How to prove $\sum_{n\ge 0} a_n x^n = \sum_{n\ge 0} \left(\sum_{k=0}^{n}(-1)^{k+1} \binom{n}{k}a_k\right) \frac{x^n}{(x-1)^{n+1}}$?

Finding a transformed random variable's distribution

Rotation of functions without using matrices

Find $\sum^{30}_{k=7}(\sqrt{k - 4} - \sqrt{k - 3})$ without explicitly calculating the 24 complements

Rectifying the isophote to second order by homothetic transofrmation

Formula for Transformation of Polynomial Coefficients under Rotation

Reducing the general sextic to "cubic" form and other Tschirnhausen transformations of higher degrees

Generalization of Gauss multiplication formula for $\Gamma(jm+kn+a);j,k\in\Bbb N$?

Removing four terms from the decic using a Tschirnhausen transformation?

Problem With Transformation of Coordinates

What is Meant by "Change of Origin" in Coordinate Geometry?

Transform ratios with negative numerators

General exponential grid

Expected value of transformation of exponential PDFs

How would I scale up a set of values while preserving their original increasing order using logarithmic functions?

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