# Questions tagged [summation]

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### Inequality with interesting independent constants

Let $b_1,\dots,b_{n-1}$ be integers satisfying $0 \le b_i \le n-i$ for each $i \in [n-1]$ such that $\sum_{i=1}^{n-1} b_i = \alpha \binom{n}{2}$ where $\alpha$ is constant strictly between $0$ and $1$....
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### How to bound $\sum e^{-a n^p}$

Let $0<p<1$ and $a>0$. Then it would seem that $$\sum_1^\infty e^{-an^p}\le Ce^{-a}$$ For some constant $C(p)$ since the terms in the summation decay exponentially. However, I can't quite ...
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### Relation between sums

Let $$A=\sum_{i=1}^N a_i\\ B=\sum_{i=1}^N b_i$$ where $0<a_i,b_i<1\ \forall i$ and $N <+\infty$ Let $$a=A/N$$ $$b=B/N$$ such as $\sum_{i=1}^N a=A$ and $\sum^N_{i=1}b=B$ is there any kind of ...
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### How do I find the partial sum of the Maclaurin series for $e^x$?

In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$. Wolfram | ...
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### Proving by Induction a Combinatorial Binomial

I'm currently stuck trying to prove that for all items in the sequence $\binom{n+1}{2}$, for all $n\geq 1$, that $\binom{n+1}{2}=\sum \limits _{i=0}^ni$. My first assumption is to solve for my base ...
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### Show that $\sum_{r=1}^{n} \frac{5r+4}{r(r+1)(r+2)}=\frac{7n^2+11n}{2(n+1)(n+2)}$ [closed]

Show that: $$\sum_{r=1}^{n} \frac{5r+4}{r(r+1)(r+2)}=\frac{7n^2+11n}{2(n+1)(n+2)}$$ I'm very lost on this question. I initially tried to identify a certain progression that this function adhered to, ...
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### Why is this step required in the proof of sum of first $n$ odd numbers using the Well Ordering Principle?

I came across this question while doing $\text{6.042J}$ from MITOCW. I have a doubt in the part c, namely, why do we need to manipulate the formula in that way? Here is my solution so far to the ...