I am trying to compute the sum  $\sum_{i=0}^{\infty} \frac{i}{2^i}$  which I know should be equal to $2$, but I cannot prove it.  If I am not mistaken, it should be a arithmetico-geometric series (), hence the title.  Any help greatly appreciated!
17492
Apparently,   $$ \sum_{n = 0}^\infty \frac{n}{2^n} $$  converges to 2. I'm trying to figure out why. I've tried viewing it as a geometric series, but it's not quite a geometric series since the numerator increases by 1 every term.
113978
I am having trouble seeing how $\sum_{n=1}^{\infty}\frac{2}{(2n+1)(2n+3)}$ equals $\sum_{n=1}^{\infty}\frac{1}{2n+1}-\frac{1}{2n+3}$. I can see $\sum_{n=1}^{\infty}\frac{1}{2n+1}+\frac{1}{2n+3}$ but not $\sum_{n=1}^{\infty}\frac{1}{2n+1}-\frac{1}{2n+3}$. All help is appreciated, thanks.
49817
Lets define the following series $$S_n =\sum_{i=1}^{\infty} \frac 1 {i(i+1)(i+2)...(i+n)}.$$ I know that $S_0$ does not converge so let's suppose $n \in N$ and define $S_n$ for some $n$. We have $S_1=1$ , $S_2=\frac1 4$ , $S_3=\frac 1 {18}$ , $S_4=\frac 1{96}$ , $S_5=\frac 1{600}$ etc.. the numerator in all results is 1the pattern in denominator is $[1,4,1,96,600...]$ and   that is equal to $n*n!$ Finally I want to prove the general equality: $\sum_{i=1}^{\infty} \frac 1 {i(i+1)(i+2)...(i+n)}=\frac 1 {n*n!}$
30637
I have been trying to find the sum $\sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{2n+1}{3^n} $. After some calculation, I got here: $\frac{-6}{8}+\frac{1}{4}+8\sum\frac{k}{9^k}$. I know the result is $\frac{5}{8}$ , and I verified it with Wolfram Alpha.I saw that $\sum_{k=1}^{\infty}\frac{k}{9^k}=\frac{9}{64}$. But I don't know how to prove the last equation: $\sum_{k=1}^{\infty}\frac{k}{9^k}=\frac{9}{64}$. I hope someone could help me or show me another method to find the sum for my initial series.
38461
I am trying to calculate the following series:  $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)}$$  and I managed to reduce it to this term  $$\sum_{n=1}^{\infty}(\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2n+4})$$  And here I am stuck. I tried writing down a few partial sums but I can't see the pattern, $\frac{1}{2}-\frac{1}{2}+\frac{1}{5}+\frac{1}{4}-\frac{1}{3}+\frac{1}{8}+...$ I cant seem to find a closed formula that we can calculate for $S_n$  How should I go about solving this question
73716
How to evaluate this infinite sum?  $$\sum_{n=1}^{\infty}\frac{1}{2^n-1}$$
7372
I'm not sure if I should call this "the sum of a sum" but I can't think of another phrase for it.   Anyhow, I'm confused about excercises such as this one  $$\sum_{j=2}^{\infty} \left( \sum_{k=2}^{\infty} k^{-j} \right)$$  where I want to compute the sum. I'm thinking maybe I could rewrite it as  $$\sum_{k=2}^{\infty} \frac{1}{k^{2}} + \sum_{k=2}^{\infty} \frac{1}{k^{3}}+... $$  but I don't know where to go from there or if that is even correct thinking. (I just began my first course in linear analysis.)
113204
How could the series below be computed ? $$\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$$ It's not a series from a book, but a series I thought of many times, and I didn't manage to figure out what I should do here. I'm just curious to know if there are some known ways for approaching such a series. Thanks!
100546
Everyone knows about the classic  $$ \sum_{i=1}^{\infty} \dfrac{1}{2^i} = 1 $$ However, is there any way to find $$ \sum_{i=0}^{\infty} \dfrac{1}{2^{2^i}} = \dfrac12 + \dfrac14 + \dfrac{1}{16} + \dfrac{1}{256} + \cdots $$
45468
How to compute the sum of the following series. I have no idea...  The series is $$\sum_{n=1}^\infty(1+1/2+\cdots+1/n)x^n.$$
28868
What would be the sum of the following series?  $$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$  Thanks
62940
Assume that $0 &lt; a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that  $$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} &lt; \infty $$ ?  I think it is but I can't prove it. Of course if $a_n \geq \varepsilon$ for some $\varepsilon &gt; 0$ this is obvious.  Any idea? Thanks.
86275
I am struggling a bit with power series at the moment, and I don't quite understand what this question is asking me to do? Am I meant to form a power series from these, or simply evaluate that series? Any explanation/working is appreciated.  Using power series representation, calculate  $$\sum_{n=1}^\infty \frac{n2^n}{3^n}.$$
76270
If we knew $\sum_{i=1}^{n} f(i)=S$  ($n$ can be $\infty$), is there anything we could say about $\sum_{i=1}^{n} \dfrac {1}{f(i)}$ in terms of $S$?  The only thing I was able to find on the web is . If there is no known relationship, is it possible for there to exist one that we just haven't discovered?
16813
I would like to know: How come that  $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$  Why isn't it infinity?
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 $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{2}$
91006
$$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$$  The series converge?  Please verify my solution below