# Tagged Questions

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### Rearranging summation terms including a complex exponential expression

I'm reading a paper on signal processing and having a hard time wrapping my head around a step the author takes. The signal of interest is defined as $r_k = e^{j(2\pi\Delta f k T_s + \theta)} + v_k$ ...
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### Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$ [duplicate]

Combinatorial argument for $$1+\sum\limits_{r=1}^{r=n} \ r\cdot r! = (n+1)!$$ The algebraic proof is easy as $r=(r+1)-1$.
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### Sum of the reciprocal of the prime-position primes.

The primes are $2, 3, 5, 7, 11, 13...$ The sum of the reciprocals of the primes diverges, proven by Euler: $$\sum_{n=1}^\infty{\frac{1}{p_n}}=\infty$$ Here, $p_n$ is the $n$-th prime. I'm asked to ...
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### Sum notation $\sum_{\sigma\in\{\pm 1\}^n}$?

I would like to know what the following sum notation means: $$\sum_{\sigma\in\{\pm 1\}^n}\left(\prod_{1\leq i\leq n}F(x_i^{\sigma_i})\right)$$ where $n$ is a positive integer, $x_i$ are some ...
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### Pull constant out of a summation of fractions

General problem $$\sum_{i=1}^n \frac{a_i + x}{b_i + x} = 0$$ Is it possible for solve for $x$? Some context I've hit a road block in my derivation... At this point, I need to pull the model ...
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### Changing summation in a power series

I'm doing a question in my power series unit that involves adding summations together, I just started this unit so I'm not totally clear on how changing summation works, from what I understand you ...
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### Find explicit formula for summation

I have this summation: $\displaystyle\sum_{i=1}^{\log_2 n} 2^{i}$, any suggestion of how get an explicit formula?
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### Summation notational convention

Please correct improper notation/terminology $$\sum_{k=0}^{n-1} ar^k$$. $$\sum_{k=1}^{n} ar^{k-1}$$ As far as I can tell these both represent the same thing. It's the partial sum {$S_n$} where the ...
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### Last Digit of $x^0 + x^1 + x^2 + \cdots + x^{p-1} + x^p$

Given $x$ and $p$. Find the last digit of $x^0 + x^1 + x^2 + \cdots + x^{p-1} + x^p$ I need a general formula. I can find that the sum is equal to $\dfrac{x^{p+1}-1}{x-1}$ But how to find the ...
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### Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$

In an interesting contest question I recently encountered, I chanced upon a question I couldn't solve. $$\sum^{2016}_{i=1}i^{101}$$ is divisible by: (a)2014 (b)2015 (c)2016 (d)2017 How would I ...
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### Sum of Gamma functions (product pairs)

This is my first time asking a question on stackexchange. Is there an analytical expression for the following summation of Gamma functions? $\sum_{t=0}^m \Gamma (A + t) \Gamma (B + m -t) = ?$ for ...
### Find the sum of the series: $\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$?
$$\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$$ I made a series to get $$\sum_{n=0}^\inf \frac{(-1)^n}{(1+2n)*2^{1+2n}}$$ but what series can it manipulate and simplify ...
### Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?
Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent? I tried ratio test but didn't seem to work, and I also know that the limit goes to zero, but I can't say its convergence because then....