Mathematics Stack Exchange
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.

17770 questions
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Find x so that summation equal to 1

I am trying to find the value of x so that this equation is true: $$x = \frac{1}{\sum_{i=1}^{p} \dfrac{e^{-p} p^{i}}{i!}}$$ Another condition is that $$\frac{x e^{-p}p^{i}}{i!}$$ Should be between 0 and 1 (inclusive). I have tried some things but I…
Nikos Koukos
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Calculating $\sum_{n=1}^\infty {\frac{1}{2^nn(3n-1)}}$

I'd appreciate any help, I know it has something to do with the geometric series but I still can't figure out how. I thought about integration but couldn't find a way to do it. $$\sum_{n=1}^\infty \frac{1}{2^nn(3n-1)}$$ Thanks.
Meno11
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Summation with a variable as the upper limit

$$\sum_{n=1}^m \frac{n \cdot n! \cdot \binom{m}{n}}{m^n} = ?$$ My attempts on the problem: I tried writing out the summation. $$1+\frac{2(m+1)}{m} + \frac{3(m-1)(m-2)}{m^2} + \cdots + \dfrac{m\cdot m!}{m^m}$$ I saw that the ratio between each of…
Jacob Copland
  • 27
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What is the value of this infinite sum?

Take the sum $\sum_1^{\infty} \frac{1}{2^n -1}$. I plugged it into Wolfram alpha, and it converges to a value around $1.606 \dots$ but wolfram didn't spit out any nice closed form. Is there an exact value for this sum?
Rob
  • 6,727
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Summation simplification help

Hi, this is an answer to a sum simplication. However, I understand the highlighted. How did the sum all of a sudden change from index 3 to index 1? I do realize that in line 2, the difference in summation is equivalent to the Left side hand side.…
Kapooky Handy
  • 595
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Sum of falling factorials

How do I find the sum with given $\alpha$: $$\sum_{k=1}^{n}\frac{\alpha^{\underline{k}}}{k}=?$$ There's a tip in the task to find $\Delta \alpha^{\underline{n}}=\alpha^{\underline{n+1}}-\alpha^{\underline{n}}$. So we get $$\Delta…
barcarolla
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Why is $1+\sum\limits_{k=0}^{j-1}\frac{1\cdot2\cdot3\dots k}{3\cdot4\cdot\dots k+2}=3-\frac{2}{j+1}$

Why is $1+\sum\limits_{k=0}^{j-1}\frac{1\cdot2\cdot3\dots k}{3\cdot4\cdot\dots k+2}=3-\frac{2}{j+1}$ If one has the result it is not difficult to verify it by induction, but how can I solve it without induction ?
ketum
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Understanding a formula

I don't get this formula: $$\sum_{i=1}^n i^2 = n(n+1)(2n+1)/6$$ Of course, I know that n means that we can plug in its place any natural number and doing the computation. Instead, I am not sure if I get what is the role of i, in this case, as the…
Always learning
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Finding the value of a summation

Question: For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1)=1$, $c(2n)=c(n)$, and $c(2n+1)=(-1)^nc(n)$. Find the value of $\displaystyle\sum_{n=1}^{2013}c(n)c(n+2)$. To solve this question, I tried to figure out…
Caddy Heron
  • 1,249
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Evaluating a summation $\sum_{k=0}^{100} k^2 + k$

I need help evaluating this: $$\sum_{k=0}^{100} k^2 + k$$ I'm not sure how to solve this without writing the equation 100 times. and then adding them up.
Joshua Mathews
  • 103
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Changing signs to minus signs to obtain a sum of zero.

Consider the sum $1+2+3+...+101$. Is it possible to change some of the plus signs to minus signs so that the sum is zero? Well, I know by using Gauss' method $1+2+...+100=5050$ then $5050+101=5151$ So I started to see if I can find a pattern but…
Caddy Heron
  • 1,249
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How to prove that $2-1+3-2+4-3+…+(n+1) - n = -1 + (n+1)$?

How can I prove that $2-1+3-2+4-3+…+(n+1) - n = -1 + (n + 1)$? I know that $2$ and $-2$ cancel out, and so do $3$ and $-3$, $4$ and $-4$, and so on, but is there a way I can prove that in such a sum, every term is cancelled out except the second…
Léo Lam
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Symmetry of Euler-Frobenius coefficient

The coefficients of an Euler-Frobenius polynomial is given by $$b_k^n=\sum_{\ell=1}^k(-1)^{k-\ell}\ell^n\left(\begin{matrix}n+1\\k-\ell\end{matrix}\right)$$ The symmetry property of this coefficients say that $b^n_k=b^n_{n+1-k}$ I've tried quite a…
PID
  • 23
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The sum of $\frac{k^{2}}{k-1}$ from 6 to 12

Can you evaluate this sum by using the properties of the sigma notation ? Or I must develop this and evaluate them one by one ?
user108343
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4 answers

Prove that $1.49<\sum_{k=1}^{99}\frac{1}{k^2}<1.99$

It can be proven by induction that $$\sum_{k=1}^{n}\frac{1}{k^2}\leq2-\frac{1}{n}$$ From here, we can easily acquire the upper bound of the sum $$\sum_{k=1}^{99}\frac{1}{k^2}$$ letting $n=100$. However, I am not quite sure about the lower bound. The…
Trogdor
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