# Tagged Questions

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### Can I know the value of an infinite serie?

$$\sum\limits_{n=0}^{\infty}\frac{n}{e^n}$$ I have found through a software that the value is $\dfrac{e}{(e-1)^2}$. I've been trying to do it manually but I am getting $\dfrac{\infty}{\infty}$, ...
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### What is the infinite sum of $a^{b^x}$?

What would $$\sum^{\infty}_{n=0}(1/2)^{4^n}$$ be and how to determine it? EDIT: Apologies. I can see this converges by the ratio test. My issue is working out its sum, more for fun really. It ...
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### Summation evaluation of $\sum_{j=0}^k n^{1/2^j}$?

How do I go about solving this: $\sum_{j=0}^k n^{1/2^j}$ So, the terms of this series are $n , n^{1/2},n^{1/4},n^{1/8},.......n^{1/2^k}$ Any insights on what the thought process should be, to ...
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### Convergence of $\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n$

We know that $\sum_{ k\in\mathbb{N} } \frac{\lambda^k}{k!} = e^\lambda$. I'm interested in the convergence of $$S^{(n)}=\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n$$ for some value ...
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Can anyone help me understand how to solve these two series? More than the solution I'm interested in understanding which process I should follow. Series 1: $$\sum_{i = 3}^{\infty} i * a^{i-1}, ... 0answers 79 views ### Upper Bound on \frac{1}{1-\beta u}-\sum\limits_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))} Is there any procedure to find an upper bound of the following expression?$$\frac{1}{1-\beta u}-\sum_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$$Here u,\beta\in\mathbb{R},\ u>1,\ ... 3answers 124 views ### Need to check simplification of expression with infinite sum of exponentials In reviewing a paper, I've come across a simplification the looks fishy to me, but I'm having a hard time checking it. I pulled out my old CRC handbook, but neither that nor Google are proving to be ... 1answer 449 views ### Use Cauchy's Multiplication Theorem and the Binomial Theorem to prove \exp(x+y)=\exp(x)\exp(y) I am to use Cauchy's Multiplication Theorem and the Binomial Theorem in order to prove \exp(x+y)=\exp(x)\exp(y)  but I have no idea where to begin. All I can think of doing is setting \exp(x) ... 1answer 94 views ### Finding a general coefficient in the multiplication of the two series Help me please to find a general coefficient a_j of the following series$$ ...
What is the appropriate way to simplify such an expression. i am unsure of how to use the series i know to apply to this situation $$\sum_{L=0}^{M}s^{L}L^{2}$$ do i modify such a series as power ...
Let's consider the sequence of real numbers $(a_{n}),n\geq1,a_{1}>0$ that satisfies the following recurrence: $$\frac{n(n+2)}{(n+1)}a_{n}-\frac{n^2-1}{n}a_{n+1}=n(n+1)a_{n}a_{n+1}$$ I'm supposed ...