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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 $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{... 10answers 981 views ### Why does$\sum_{n = 0}^\infty \frac{n}{2^n}$converge to 2? [duplicate] 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 ... 6answers 931 views ###$\sum \limits_{n=1}^{\infty}n(\frac{2}{3})^n$Evalute Sum [duplicate] Possible Duplicate: How can I evaluate$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$How can you compute the limit of$\sum \limits_{n=1}^{\infty} n(2/3)^n$Evidently it is equal to 6 by wolfram alpha ... 4answers 19k views ### How to calculate:$\sum_{n=1}^{\infty} n a^n$[duplicate] I've tried to calculate this sum: $$\sum_{n=1}^{\infty} n a^n$$ The point of this is to try to work out the "mean" term in an exponentially decaying average. I've done the following: $$\text{let }... 2answers 53k views ### Sum of a power series n x^n [duplicate] I would like to know: How come that$$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$Why isn't it infinity? 6answers 1k views ### Sequence sum question: \sum_{n=0}^{\infty}nk^n [duplicate] I am very confused about how to compute$$\sum_{n=0}^{\infty}nk^n.$$Can anybody help me? 5answers 10k views ### Compute 1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \cdots + n \cdot \frac {1}{2^n} + \cdots [duplicate] I have tried to compute the first few terms to try to find a pattern but I got$$\frac{1}{2}+\frac{1}{2}+\frac{3}{8}+\frac{4}{16}+\frac{5}{32}+\frac{6}{64}$$but I still don't see any obvious ... 6answers 215 views ### Sum of infinite series 1+\frac22+\frac3{2^2}+\frac4{2^3}+\cdots [duplicate] How do I find the sum of \displaystyle 1+{2\over2} + {3\over2^2} + {4\over2^3} +\cdots I know the sum is \sum_{n=0}^\infty (\frac{n+1}{2^n}) and the common ratio is (n+2)\over2(n+1) but i ... 5answers 4k views ### Formula for r+2r^2+3r^3+…+nr^n [duplicate] Is there a formula to get r+2r^2+3r^3+\dots+nr^n provided that |r|<1? This seems like the geometric "sum" r+r^2+\dots+r^n so I guess that we have to use some kind of trick to get it, but I ... 4answers 216 views ### Question on a tricky Arithmo-Geometric Progression:: [duplicate]$$\dfrac{1}{4}+\dfrac{2}{8}+\dfrac{3}{16}+\dfrac{4}{32}+\dfrac{5}{64}+\cdots\infty$$This summation was irritating me from the start,I don't know how to attempt this ,tried unsuccessful attempts ... 5answers 4k views ### What is the sum of the series 1/3 + 2/9 + 3/27 + 4/81 + … [duplicate] I remember solving this in highschool , but now I don't remember how to find sum of these kind of series . I want to find the sum of the general series Sum \sum_{n=1}^{\infty} n .a^{-n} = ? ... 2answers 5k views ### Find value of infinite sum [duplicate] Possible Duplicate: How can I evaluate \sum_{n=1}^\infty \frac{2n}{3^{n+1}} How would I go about deriving the value of the following infinite sum: \sum\limits_{k=1}^\infty kx^k ? I thought ... 3answers 388 views ### Evaluating the series \sum\limits_{n=0}^{\infty}\frac{n}{3^n} [duplicate] Possible Duplicate: How to find the sum of this infinite series Hello all, I have one last major question, where would I get started on the following question:$$\sum_{n=0}^{\infty}\frac{n}{3^n}$...
So I am having trouble getting the sum of the series: $1 + 2\left(\frac32\right) + 3\left(\frac{3^2}{2^2}\right) + ... + k\left(\frac{3^{k-1}}{2^{k-1}}\right)$ I cant figure out for the life of me ...
### Evaluate the series $\sum _{n=1}^{\infty} \frac{n}{5^n}$ [duplicate]
$$\sum _{n=1}^{\infty}\frac{n}{5^n}$$ I tried to plug in $n=1,2,3,4,...$ but I can't use common ratio to solve problem. I think there is another way like using differentiation or integral but I don't ...