Questions tagged [alternating-expression]

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Proof of the Laplace formula using alternating tensors.

I am reading "Differential Forms with applications to the physical sciences" of Harley Flanders. I came across the proof of the Laplace formula, there were some details left to the reader, I ...
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2 answers
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Alternating sum of binomial coefficients $\sum_{k=0}^{49}(-1)^k \binom{99}{2k} = -2^{49}$

How can you prove that $$\sum_{k=0}^{49} \binom{99}{2k}(-1)^k = -2^{49}?$$ A more general formula seems to be $\sum_{k=0}^{n} \binom{2n + 1}{2k}(-1)^k$ For $n = 0:$ it equals $2^0$ For $n = 1: -2^1$ ...
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2 answers
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Closed form of $\sum _{k\ge 1} \frac{(-1)^{\binom{k}{p}}}{k}$, an alternating harmonic series with the signs determined by a binomial coeffcient

In a comment to Evaluating $\int_{0}^{1} \lim_{n \rightarrow \infty} \sum_{k=1}^{4n-2}(-1)^\frac{k^2+k+2}{2} x^{2k-1} dx$ for $n \in \mathbb{N}$ I proposed to study this alternating harmonic sum $$s(p)...
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1 answer
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Evaluate a kronecker symbol sum: $\sum\limits_{n=1}^\infty \frac {\big(\frac n x\big)}n$ and $\sum\limits_{n=1}^\infty \frac {\big(\frac xn\big)}n$

The Kronecker Symbol $\left(\frac nm\right)$ has a range of $\{-1,0,1\}$ and $\sum\limits_{n=1}^\infty\frac{(-1)^n}n=-\ln(2)$, so we combine to find the following with the using software. Also note ...
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1 vote
6 answers
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Proving $(ac+bd)^2+(ad-bc)^2=(a^2+b^2)(c^2+d^2)$ with various solutions.

$(ac+bd)^2+(ad-bc)^2=(a^2+b^2)(c^2+d^2)$ Solutions in the answers. $\ \\ \ \\ \ \\ \ \\$ Edit) Since this question is closed, I'll add more contexts for this question. This identity is called "...
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