# Questions tagged [summation]

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### Is there an easy way to calculate this infinite summation

Is there an easy way to calculate this summation of integral: $$\sum_{n=0}^\infty \int_{r=0}^1 \frac {(r-\frac{1}{2})\cos(c\cdot\ln(r+n))} {(r+n)^{1-b}} dr$$ The most obvious approach is to calculate ...
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### Simplify Summation $S_n=\sum_{k=1}^{8n} (-1)^\frac{(k)(k+1)(k+2)}{6}(k)^2+\sum_{k=1}^{8n}(-1)^\frac{(k+2)(k+3)}{2} (k)^2-4\sum_{k=1}^{8n}(8k-2)^2$

If $$S_n = \sum_{k=1}^{8n} (-1)^\frac{(k)(k+1)(k+2)}{6} (k)^2 + \sum_{k=1}^{8n} (-1)^\frac{(k+2)(k+3)}{2} (k)^2 -4\sum_{k=1}^{8n} (8k-2)^2$$ then the value of $-S_{40}$ is equal to? Simplifying all 3 ...
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### Directional cosine of normal unit vector

In below directional cosine of binormal unit vector instead of $${l} = {y'z''-z'y''}/ \sqrt{(y'z''-z'y'')^2+(y'x''-x'z'')^2+(y'z''-z'y'')^2}$$ and similarly m and n DC. Following term are used using ...
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### Empty sum of undefined function

An undergraduate real analysis homework problem I am working on raised the following question: Does function $f$ need to be defined for $i=0,1$ for empty sum $\sum_{i=1}^0{f(i)}$ to be equal to zero? ...
1 vote
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### Trying to understand this skipping sum

$$f(x) = \begin{cases} 0 & x \not\equiv 4 \pmod 5 \\ 1 & x \equiv 4 \pmod 5 \end{cases}$$ and $$f(x) = \frac{1}{5} \sum_{k=0}^4 \cos\left(\frac{2 \pi}{5} k (x-4) \right)$$ Can someone help ...
1 vote
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### Asymptotic equivalent of $\sum_{k=1}^n a^k k^{-1/2}$

I encountered recently the following partial sum $\sum_{k=1}^n a^k k^{-1/2}$ with $a$ a constant approximately equal to $2.955$. I was wondering if there were any clever way to find an asymptotic to ...
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### Is $\lim_{m\to\infty}\sum_{k≥t}\frac{1}{\binom kt^m}=1$?

Context: Using Wolfram calculator, I've observed that : $$\sum_{k≥2}\frac{1}{\binom k2^{100}}≈1$$ $$\sum_{k≥5}\frac{1}{\binom k5^{100}}≈1$$ $$\sum_{k≥4}\frac{1}{\binom k4^{50}}≈1$$ Question: I want to ...
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### Closed form solution for partial summation of $\sum_{x=1}^{k} \frac{2^{\frac{1}{x}}}{{x^2}}$

Recently I've been working on solving summations and I found this one to be quite tricky. $\sum_{x=1}^{k} \frac{2^{\frac{1}{x}}}{{x^2}}$ The integral which this is based off of, can be solved with u ...