# Questions tagged [summation]

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### How do we get the result of the summation $\sum\limits_{k=1}^n k \cdot 2^k$? [duplicate]

Possible Duplicate: Formula for calculating $\sum_{n=0}^{m}nr^n$ Can someone explain step by step how to derive the following identity? $$\sum_{k=1}^{n} k \cdot 2^k = 2(n \cdot 2^n - 2^n + 1)$$
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### Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
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### How to prove a formula for the sum of powers of $2$ by induction?

How do I prove this by induction? Prove that for every natural number n, $2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$ Here is my attempt. Base Case: let $n = 0$ Then, $2^{0+1} - 1 = 1$ Which is true. ...
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### Evaluate $\sum\limits_{k=1}^n k^2$ and $\sum\limits_{k=1}^n k(k+1)$ combinatorially

$$\text{Evaluate } \sum_{k=1}^n k^2 \text{ and } \sum_{k=1}^{n}k(k+1) \text{ combinatorially.}$$ For the first one, I was able to express $k^2$ in terms of the binomial coefficients by considering a ...
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### Reference for a tangent squared sum identity

Can anyone help me find a formal reference for the following identity about the summation of squared tangent function: $$\sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+.$$ I ...
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### Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots$

Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots$$ I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a ...
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### Solve summation $\sum_{i=1}^n \lfloor e\cdot i \rfloor$

How to solve $$\sum_{i=1}^n \lfloor e\cdot i \rfloor$$ For a given $n$. For example, if $n=3$, then the answer is $15$, and it's doable by hand. But for larger $n$ (Such as $10^{1000}$) it gets ...
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### Geometrical interpretation of $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$

Using induction it is straight forward to show $$\left(\sum_{k=1}^n k\right)^2=\sum_{k=1}^n k^3.$$ But is there also a geometrical interpretation that "proves" this fact? By just looking at those ...
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### Where do summation formulas come from?

It's a classic problem in an introductory proof course to prove that $\sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2}$ by induction. The problem with induction is that you can't prove what the sum is ...
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### A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$

So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling ...
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### How do I evaluate this sum(involving the floor function)? [duplicate]

$$\sum_{i=1}^N\left\lfloor\frac{N}{i}\right\rfloor$$ Is there a closed form expression to the above sum? (Mathematica doesn't give me anything)
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### Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?

How can we evaluate this series $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$ Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how ...
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### Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$

Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$; I was stuck with this question for a while... Help me please!!! Thanks!!!
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### How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate]

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have ...
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### Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$

I am trying to prove this binomial statement: For $a \in \mathbb{C}$ and $k \in \mathbb{N_0}$, $\sum_{j=0}^{k} {a+j \choose j} = {a+k+1 \choose k}.$ I am stuck where and how to start. My steps ...
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### Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$
How does one sum the series $$S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots$$ This was asked to me by a high school ...
### Show that $\sum\limits_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\frac{1}{p!p}$ for every positive integer $p$
I have to prove that $$\sum_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\dfrac{1}{p!p}$$ How can I do that?