For questions on exponential sums, such as $\sum \exp(2\pi ix_n)$.

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-1
votes
1answer
94 views

Equation to the summation to $ 3\cdot 10^{-n-1}$

What would the equation to sum (shown below) of $ 3\cdot 10^{-n-1}$? Just like $ 2^x-1$ is the sum of $ 2^{x-1}$. $$\sum_{n=0}^{\infty} 3\cdot 10^{-n-1}$$ This would be 0.3+0.03+0.003. . . this ...
5
votes
1answer
386 views

Integrating $\sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$

Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$ Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
4
votes
2answers
424 views

Baker-Hausdorff Lemma from Sakurai's book

I'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + ...
4
votes
1answer
202 views

Determining the Value of a Gauss Sum.

Can we evaluate the exact form of $$g\left(k,n\right)=\sum_{r=0}^{n-1}\exp\left(2\pi i\frac{r^{2}k}{n}\right) $$ for general $k$ and $n$? For $k=1$, on MathWorld we have that ...
3
votes
3answers
231 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 ...
1
vote
2answers
83 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}, ...