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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The value of $\sum _{n=1}^{\infty }{\left(q\right)^n\left(\sin(na)\right), |q|<1}$

I need to solve this sum, but unfortunately I have no idea how to start.I would be grateful for every advice. $$\sum _{n=1}^{\infty }{\left(q\right)^n\left(\sin(na)\right), |q|<1}$$
Many
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Sum of Fourth Powers of cosine series has closed form solution.

A problem posed in the 1988 Irish Mathematical Olympiad asks for a proof of the following $$\sum\limits_{k=1}^{n} \cos^{4}\Big(\frac{k\pi}{2n+1}\big) = \frac{6n-5}{16}$$ Can anyone give me a heads-up on how to proceed to prove this very interesting…
Callie12
  • 581
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A Double sum with something weird between the summations

$$\sum_{k=1}^{\infty} \Bigg[ \dfrac{(-1)^{k-1}}{k} \sum_{n=0}^{\infty} \dfrac{300}{k \cdot 2^n +5}\Bigg]= ?$$ I've never seen such a complex summation before. Can anyone help me? It would also be extremely helpful if anyone could tell me if they've…
Lee Laindingold
  • 421
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How was the index replace?

This is page from The Art of Computer Programming by D.E.Knuth. I am interested in circled and underlined Why he is just change $n-j$ on $j$ ?
Mouvre
  • 193
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Simplify formula by removing sum from it

I have a question about the following formula, I just need to write it without a sum with some formula convertion(simplify it). As for now I'm stuck, I don't know if it is possible for now, but hope it is. $$ \sum_{i=1}^l \frac{b_i^2}{1+AB_i}, \quad…
Donvino
  • 83
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Simplifying the summation $ (3^i\sqrt{n/3^i})/\log_2(n/3^i)$ from $i=0 \text{ to }k-1$

I have been trying to simplify the following summation with the intention of breaking it into less complex summations, but I keep getting stuck no matter what I try: $$\sum_{i=0}^{k-1} 3^{i} \cdot…
MikeKatz45
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When Sigma-summation stops with a decimal value

$\displaystyle \sum_{n=1}^{\log_2x}$... When you have a sum like this, what is the result of $\log2_x$ since it may return a decimal/fractional value? Is it truncated, ceiled or floored? I know I could do: $\displaystyle…
user366820
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Finding sum of a finite series.

Consider the series $$\frac {q_1} {p_1} + \frac {q_1 q_2} {p_1 p_2} + \cdots + \frac {q_1q_2 \cdots q_n} {p_1 p_2 \cdots p_n}$$ where $p_i + q_i = 1$ and $0 < p_i < 1$ and $0 < q_i < 1$ for all $i=1,2, \cdots , n$. How can I find the sum of this…
little o
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summation of $\sum_{n=1}^\infty \frac {2^n+4^n}{6^n}$

So I need to show that this sum converges to 4.5. But when i did this is got the sum converges to 2.5. $$\sum_{n=1}^\infty \frac {2^n+4^n}{6^n}$$ My workings: $$\sum_{n=1}^\infty \frac {2^n+4^n}{6^n}=\sum_{n=1}^\infty \frac…
H.Linkhorn
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Compute $\sum_{k=1}^{n} \frac 1 {k(k + 1)} $

More specifically, I'm supposed to compute $\displaystyle\sum_{k=1}^{n} \frac 1 {k(k + 1)} $ by using the equality $\frac 1 {k(k + 1)} = \frac 1 k - \frac 1 {k + 1}$ and the problem before which just says that, $\displaystyle\sum_{j=1}^{n} a_j -…
papercuts
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Is this some magical summation?

I have been playing around with some probability calculations and somehow came to this expression $$\sum_{u_1 = 1}^n \sum_{u_2 = 1}^n \ldots \sum_{u_n = 1}^n \frac{1/u_1}{\sum_{j=1}^n 1/u_j} \prod_{i=1}^n…
Abas
  • 355
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Sum of $\sum_{n=1}^{\infty} \frac {x^n}{(n-1)!} $

the task is to find sum of $\sum_{n=1}^{\infty} \frac {x^n}{(n-1)!} $. I've tried to factor out $x$ to get $x\sum_{n=1}^{\infty} \frac {{x}^{n-1}}{(n-1)!}$ and integrate the sum, which yields $x\sum_{n=1}^{\infty} \frac {x^n}{n!} $ which I can sum…
Tomáš Svoboda
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Simplify $\sum_{i=1}^n(\sum_{j=i}^n j)$

$\sum_{i=1}^n(\sum_{j=i}^n j)$ This really is lame, but i couldn't figure out how to work with this one. I can easily tell that $\sum_{i=1}^ni= \dfrac{n(n+1)}{2}$, and that the summation i am trying to simplify should be something like -…
Yariv Levy
  • 1,145
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How can I prove this odd property?

I have recently noticed the odd fact that $$123456789\times8+9=987654321$$ So, I started looking into this and I couldn't find anything on it. I decided to figure out a formula for this and what I came up with is…
Tbw
  • 995
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Double summation identity

I'm trying to understand the following identity from here $$\sum_{k\le j \le i\le n} a_{i,j} = \sum_{i=k}^n\sum_{j=k}^i a_{i,j} = \sum_{j=k}^n\sum_{i=j}^n a_{i,j} = \sum_{j=0}^{n-k}\sum_{i=k}^{n-j} a_{i+j,i}$$ There seems to be more than just index…
mdcq
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