# Questions tagged [summation]

2,001 questions
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### Summation $k=n-4$ to $k=4$ of $1$

This is perhaps very basic but I am currently very lost on how to think in order to end up with the answer: $9-n$ in the following summation: $$\sum_{i=n-4}^4 1 = 9-n$$ My first idea was to ...
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### Calculate the following sum

Calculate the following sum $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)}$$ I Tried decomposing the denominator but still seem to be stuck thanks to the numerator
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### Basel Problem - Area of $\frac 16$ of Circle with Radius $\sqrt{\pi}$.

There are several proofs to the solution of the well-known Basel Problem, i.e. $$\sum_{n=1}^\infty \frac 1{n^2}=\frac {\pi^2}6$$ Is is possible to create a geometrical interpretation of this ...
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### Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
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### Interpretation of sums using $\cdots$

Consider the sum $$\sum_{1\le k_1 < k_2 < \cdots < k_r \le n}k_1k_2\ldots k_r$$ Does this simply mean $$\sum_{\substack{|K|=r}\\\inf(K)\ge 1\\\sup(K)\le n}\prod_{k\in K} k$$ I am ...
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### Discrete Math-Computing Summations

So I'm asked to compute a summation with an upper limit $k = 20$ and lower limit $k=1$, where: $B_k= 0$ when $k=1$, and $B_k = \dfrac{1}{(k^2-1)}$ , for $k>1$. I was wondering if there is a ...
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### Average length of a bitstring

I am trying to compute the average length of a bit string from all bit strings of $\{0,1\}^n$. By length n I mean a bit string of length n where the most significant bit is 1. I know there is $2^0$ ...
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### Compound interest with a compounding interest rate

I have an investment which pays 3% interest (r) annually but it also increases the interest rate every year by 5% (g). I re-invest all interest payments at the start of each year. How many years (t) ...
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Supposing I have the following sequence based on two indexes: $a$ and $b$. For $a$ starting with $1$ and $b$ starting with $5$ we have the following sum: $$1 \cdot 5 + 2 \cdot 4 + 3 \cdot 3 + 4 \... 2answers 49 views ### How should I solve this summation problem? Lets say that we have these x and y coordinates x=1,2,3,4,5 and y=6,7,8,9,10 and where n=5. How would I use these x coordinates with the first summation? Now, I know that learning is ... 2answers 52 views ### Inequation of an sum smaller than 1 I'm trying to figure out the following$$ \sum^{\infty}_{n=3} \dfrac{q!^2}{n!^2} < 1 $$How I can show it if q \geq 2? Maybe with telescoping sums? Thanks, Landau 2answers 48 views ### Explanation for sum of sequence I saw that in a textbook. Could somebody explain how this sum of a sequence was obtained? ⌈n/2⌉+...+⌈n/2⌉+⌈n/2⌉ = ⌈(n+1)/2⌉⌈n/2⌉ OP has ... 2answers 186 views ### Compute \sum_{i=0}^{2n} (-3)^i by splitting the series into two parts. Compute \sum_{i=0}^{2n} (-3)^i by splitting the series into two parts. How do I split it into two parts? All I can tell so far is that the sum is going to be a positive number (probably) because ... 2answers 144 views ### finite sum over a Gaussian I have a sum of the form:$$\sum_{n,m=-N}^N e^{-\alpha (n-m)^2}$$where \alpha > 0 is some constant, and n,m take the integer values: -N,..,N. I know there is a possibility of exchanging ... 2answers 963 views ### Calculating a Summation Involving 2 Variables Having not taken a math course for multiple years, I appear to have forgotten some bare basics. Unfortunately, Google has not taken me to a solid answer after much searching. How do you solve an ... 2answers 124 views ### Simple addition of summation I know the following equality holds from previous work \sum_{i=1}^{n}a_i^2\sum_{j=1}^{n}b_j^2 + \sum_{i=1}^{n}b_i^2 \sum_{j=1}^{n}a_j^2 = 2(\sum_{i=1}^{n}a_i^2)( \sum_{i=1}^{n}b_i^2) But when I ... 2answers 45 views ### What is the difference between the summations? What is the difference between the summation$$\sum_{1 \leq i<j \leq n} f(i,j)$$and$$\sum_{1\leq i} \sum_{<j \leq n} f(i,j)?$$2answers 244 views ### Calculating double sum How do I calculate this sum (as a function of n)? I have no experience in calculating sums like these, so I don't know any of the rules regarding this subject. 2answers 48 views ### Multiplication problem I have a very simple question As 0.999⋯≠1 Hence, ... 2answers 172 views ### Finding a closed form solution For the sequence 0 2 8 34 144 ... The recurrence relation is:$$\begin{align*} E(n) = 4*E(n-1)+E(n-2) \end{align*} How to calculate the closed form expression ...
If limit exists, then what is its value? And if it does not exist then can we find where does this expression tends as $n \to \infty$. The expression : \$\lim\limits_{n \to \infty } \sum\limits_{k=...