3
votes
3answers
195 views

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 ...
2
votes
1answer
55 views

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 ...
1
vote
2answers
80 views

How to solve these series?

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}, ...
2
votes
0answers
71 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,\ ...
2
votes
3answers
102 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 ...
1
vote
1answer
370 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 $\exp(x)$ ...
0
votes
1answer
92 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 $$ ...
1
vote
3answers
367 views

Summing a exponential 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 ...
1
vote
1answer
57 views

The limit of a sum with elements from 2 sequences

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 ...