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Mathematics Stack Exchange

Questions tagged [summation]

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

The notation $\sum\limits_{i=1}^na_i$ means $a_1+\ldots +a_n$.

Use for sums of infinite series and questions of convergence; use for questions about finite sums and simplification of expressions involving sums.

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How to Split a Sequence of Numbers Into Four (Relatively) Equal Summations

How would I go about splitting a sequence of numbers into four equal (as equal as possible) summations? Say I have a sequence of 26 integers like so: 16, 4, 17, 10, 15, 4, 4, 6, 7, 14, 9, 17, 27, 6, 1, 9, 0, 12, 20, 8, 0, 3, 4, 0, 3, 4 I want to…
mforrest
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Closed-form for rational power sum

$$s_n = \sum_{k=1}^n e^{1/k}$$ This sum came up while practicing closed-form finding on a calculus book's infinite series chapters. Using Concrete Mathematics' perturbation method, I arrive at $$s_n + e^{1/k+1} = e + \sum_{k=1}^n e^{1/k+1}$$ Which I…
Charles
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Concrete Mathematics Multiple Sum index change

In Concrete Mathematics, after the "Rocky Road" equality, the authors introduce the problem $$\sum_{1\leq j< k\leq n} \frac{1}{k - j}$$ They eventually arrive at the solution involving letting $k \to k + j$ to simplify the sum. $$\sum_{1\leq j< k +…
user246500
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Ratio of two summations

I devised this question based on recent (and not-so-recent) MSE questions on summations. Evaluate $$\frac{1\cdot2+2\cdot3+3\cdot4+\cdots+1000\cdot1001}{1\cdot1000+2\cdot999+3\cdot998+\cdots+1000\cdot1}$$ or more formally Evaluate …
Hypergeometricx
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Sums involving exponential functions

I am trying to find the closed form of the following related sums: $$(i)\quad\quad S_1(n)= \sum_{m=-\infty}^{m=\infty} |n-m| e^{-p(|n-m|+|m|)} $$ $$ (ii)\quad\quad S_2(n)= \sum_{m=-\infty}^{m=\infty} m(|n-m|+\gamma) e^{-p(|n-m|+|m|)} $$ $$ (ii)…
Nick P
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Abel's summation formula for functions depending on limit of sum

Abel's summation formula states that for two functions $f$ and $g$, with $f$ differentiable, we have $$\sum_{k=1}^n f(k)g(k)=G(n)f(n)-\int_1^n G(x)f'(x)\; dx \tag{$*$}$$ where $G(n)=\sum_{k=1}^n g(k)$ (it is possible to generalize to other bounds,…
Avi
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Mathematically Expressing the Sum of...

I noticed that $$\large{2x-1 = \frac{x}{2} + \frac{\frac{x}{2}}{2} + \frac{\frac{\frac x2}2}2} + \cdots$$ until the output of one of the steps in the pattern equals $1$. Or in other words $2x-1$ is equal to the sum of $\large{\frac{x}{2}}$ plus that…
Nick Suwyn
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Correct usage of sum sign

I want to express in a formula that variables with a certain property shall be added. I think it is best expressed by an example. $A$ is a superset of $B$ and $C$. $B$ can have the properties $X$ and $Y$. $C$ also can have the properties $X$ and…
Thomas
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Can $s\sum_{n=0}^{y}(t/s)^{n/y} \ge x$ be solved for $y$?

Is it possible to solve the following equation for y? $s\sum_{n=0}^{y}(t/s)^{n/y} \ge x$ I'm trying to write a slot machine program (for a school assignment I'm making harder than it needs to be for no good reason). When "pulled", I want each…
user239731
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Hypothetical Church Growth Equation

To preface this, it's been years since my last calc class, but I feel like this is lower level than that. I am looking for the notation for an equation that sums up the values of the same equation iterated several times, while carrying the last…
joshrathke
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What is the sum $\sum_{k=0}^{n-1} e^{kx}$?

My Precalc teacher gave me this as a question and I simply cannot figure out how to do it.
user237616
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How do I display this as a sum?

How do I write this infinite series using the sigma notation? $$1+f'(n)m+\frac{f''(n)}{2!}m^2f(n)+\frac{f'''(n)}{3!}m^3(f(n))^2+...$$ My attempt: $$\sum_{k=0}^\infty\frac{f^{(k)}(n)}{k!}m^k(f(n))^{k-1}$$ But this yields an initial value of…
Sam Houston
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Changing the limits of a summation

Sorry about bad englsih Guys, i have this: n n-1 ( Σ 3k²-k) + ( Σ 2k-3k²) k=1 k=0 So, the limits of the first one are: k=1 and n and of the second one are k=0 and n-1. To proced, i have to adjust the limits until both have…
GGirotto
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How do I find a solution to this this finite series? $ \frac{1}{n^4} \sum_{i=1}^{n} \left({i^3}\right) $

How do I find a solution to this this finite series? Any help would be greatly appreciated. $$ \frac{1}{n^4} \sum_{i=1}^{n} \left({i^3}\right) $$
sean
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Dealing with constants in summation notation

I haven't worked with summation notation in a while, and am unsure how to approach the following: $\sum_1^n [-\frac 12 * \frac{(x_i - \alpha)^2}{\alpha}]$ where $\alpha \in R$ What would be the best/correct way to simplify this, i.e. pull the…
kathystehl
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