What is $$\sum_{i=0}^{\log n}i\cdot 2^i\:?$$ I know it's $O(n\cdot\log n)$ but how do you calculate this sum?
1602
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
94431
In Calculus, how do I calculate this sum?  $$f(x) = \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^n}$$  This is what I did so far: $$ f(x) = 2x \cdot \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^{n-1}} $$  Therefore: $$ \frac{\int{f(x)}}{2x} = \sum_{n=0}^{\infty} {2^n \cdot x^n}$$  But I have no idea where to continue from here!
82277
How to prove that   $$ -2\log(2) = -2 + \sum_{n=1}^{\infty}\frac{1}{n(2n+1)} $$  I know that this sum is equal to $\phi(1/2)+\gamma$ where $\phi(x)$ is the digamma function and $\gamma$ is the Euler-Mascheroni constant, but I want to evaluate the sum without knowing it.
68440
$\displaystyle\sum_{i=0}^n2^i=2^{n+1}-1$.  I don't understand induction so I could use some help.
98452
Let $x_{1},x_{2},\cdots,x_{n}$ be real numbers. Show that     $$\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|.$$   I think this problem may be solved using nice methods, but I can't find them yet; I know this may be of use: $$|a|+|b|\ge |a+b|.$$ But I can't make it work. Thank you everyone.
40295
I'd like to see a proof of this identity I discovered:  $$ \sum_{i=1}^{n}\sum_{j=1}^{i}a_j = \sum_{i=1}^{n}a_i(n-i+1) $$  Probably there's something I forgot because I couldn't manage it.  [edit]Possibly with just identities, not examples please!  [edit2]No induction procedure, either.
34609
how can I find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $? It is so weird for me because I put this to Mathematica and it tells me that sum does not converge...  Let consider sum no to infinity, but to n  $$ \sum_{k=1}^{n} (k\cdot \ln \frac{2k+1}{2k-1} - 1) =$$ $$ ln \frac{3}{1}\cdot \left(\frac{5}{3}\right)^2 \cdot...\cdot \left(\frac{2n+1}{2n-1}\right)^n - n = ln \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1} - n $$ but $$ n = ln e^n $$ so it will be $$ln\frac{1}{e^n} \cdot \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1}$$  So the limit of it is $-\infty$ Have I done this well or I missed sth?
60551
$$(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$$  I noticed this only because $\displaystyle \sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\displaystyle \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$.  But the two things look completely different and I can't think of an intuitive reason I would have seen this connection. Something like the 'Proof Without Words' images. Or a trick I could use while expanding the right hand side to transform it to the sum of cubes.
10400
Suppose I have $$ \large\sum_{i=1}^{10} 2. $$ Do I just add $2$ to itself $10$ times? I have worked on more complex ones with $n$ and such in the place where the $2$ is, but I have never done it when there is just a number there.
9386
Can you please explain why $$ \sum_{k=1}^{\infty} \dfrac{k}{2^k} = \dfrac{1}{2} +\dfrac{ 2}{4} + \dfrac{3}{8}+ \dfrac{4}{16} +\dfrac{5}{32} + \dots = 2 $$  I know $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{2}$
23708
Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$ ?    I know that $\sum_{k=0}^{n}\binom{n}{k}= 2^{n}$ and so, $\sum_{k=1}^{n}\binom{n}{k}= 2^{n}-1$ but how to deal with $k$ ?
114387
Let $x$ be a positive integer. I have the following iterated sums:  $$f(x)=\sum_{i_{u-1}=1}^{x}\sum_{i_{u-2}=1}^{i_{u-1}} \cdots \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1$$  What is the asymptotic behavior of the total?  Thank you
19041
I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$  I know that $$2^n = \sum\limits_{k=0}^n \binom{n}{k}$$ but that didn't help me
103332
Can anyone help me with this?  What is the sum of this series:  $\frac{2}{\pi}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$  I got it after plugging $x=-1$ in a Fourier series   Thank you!
41777
Possible Duplicate:          I got this question in homework:  Find an expression for the sum ‫‪     $\sum k = 1 +\cdots + n‬‬$.    and prove it using an induction.   I'm not even near finding the expression. What I did notice is that if $n$ is (for example) 5 then the sum would be   $5^2 - 4^2 + 3^2 - 2^2 + 1^2$  So the first number is always positive and from there on the sign changes.  Any tips on how do I contintue from this point on, assuming I'm in the right direction?  Thanks!
14153
I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.  For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ according to   Thanks!
33414
Possible Duplicate:          This was part of a homework assignment that I had, and I couldn't figure it out. Now it is bugging me. Can anyone help me? Although a proof would be nice, I wouldn't mind a push in the right direction.   $\displaystyle\sum_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$