Questions tagged [summation]

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

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How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$ [duplicate]

I know that $$\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n),$$ but I cannot find a way how to prove it. I tried induction but it did not work. On wiki they say that I should use differentiation but I do ...
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Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$

My question is: Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer. I'm stuck at the basis step. If I ...
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Evaluating even binomial coefficients

Can someone give me a hint how to evaluate $$\binom{n}{0}+\binom{n}{2}+\cdots+\binom{n}{o(n)},$$ where $o(n)$ is $n$ if $n$ is even and $n-1$ otherwise ?
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Summation of binomial coefficients [duplicate]

Is there a closed formula for: $\sum_{i=1}^{N}{\binom{i+k}{i}}$ ( k is a constant whole number )
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Calculating $\sum_{k=1}^nk(k!)$ combinatorially [duplicate]

The sum $\sum_{k=1}^nk(k!)$ can be easily calculated by noting $k(k!)=(k+1)!-k!$. Is there a way to calculate the sum nicely using a combinatorial argument. Is it possible to notice it is $(n+1)!-1$ ...
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Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$ [duplicate]

Prove that for $n\geq 2, \: \sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$ I used induction and I compared the LHS and the RHS but I'm getting an incorrect inequality
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Formula for $\sum_{k=1}^n \frac{1}{k(k+1)(k+2)}$?

I was trying to find a formula for the series $$\sum_{k=1}^n \frac{1}{k(k+1)(k+2)} =?$$ I tried to break this into partial fractions...To see if I could telescope this series.. The partial ...
638 views

For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$

For the Fibonacci sequence $F_1=F_2=1$, $F_{n+2}=F_n+F_{n+1}$, prove that $$\sum_{i=1}^n F_i= F_{n+2} - 1$$ for $n\ge 1$. I don't quite know how to approach this problem. Can someone help and ...
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Partial Sums of Geometric Series

This may be a simple question, but I was slightly confused. I was looking at the second line $S_n(x)=1-x^{n+1}/(1-x)$. I was confused how they derived this. I know the infinite sum of a geometric ...
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Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction [closed]

I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing. $$\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$$ Please help!
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Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k$ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear ...
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How prove this $\sum_{k=0}^{n} \frac{\binom{2n-k}{n}}{2^{2n-k}}=1$

Show that $$\sum_{k=0}^{n}\dfrac{\binom{2n-k}{n}}{2^{2n-k}}=1$$ I think this problem can be solved with nice methods, such as algebraic ones. Or can I use probability methods? Thank you
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Prove $\sum\limits_{j=1}^k(-1)^{k-j}j^k\binom{k}{j}=k!$ [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
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Prove that $\frac{1}{4-\sec^{2}(2\pi/7)} + \frac{1}{4-\sec^{2}(4\pi/7)} + \frac{1}{4-\sec^{2}(6\pi/7)} = 1$

How can I prove the fact $$\frac{1}{4-\sec^{2}\frac{2\pi}{7}} + \frac{1}{4-\sec^{2}\frac{4\pi}{7}} + \frac{1}{4-\sec^{2}\frac{6\pi}{7}} = 1.$$ When asked somebody told me to use the ideas of ...
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How to compute $\sum^n_{k=0}(-1)^k\binom{n}{k}k^n$

When trying to answer this question I arrived at $$\int^\infty_0\frac{\sin(nx)\sin^n{x}}{x^{n+1}}dx=\frac{\pi}{2}\frac{(-1)^n}{n!}\sum^n_{k=0}(-1)^k\binom{n}{k}k^n$$ After using Wolfram Alpha to ...
327 views

Using right-hand Riemann sum to evaluate the limit of $\frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$

I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, ...
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proving $\left(1+\frac 1n\right)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem

$$\left(1+\frac 1n\right)^{n} = 1 + \sum\limits_{k=1}^n \left\{\frac 1{k!}\prod_{r=0}^{k-1}\left(1-\frac rn\right)\right\}$$ this exercise is taken from Apostol's Calculus I (page 45) and it's ...