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7

We have $$\begin{align} \sum_{i=1}^n\sum_{j=1}^i(2^j-1) &=\sum_{i=1}^n\sum_{j=1}^i2^j-\sum_{i=1}^n\sum_{j=1}^i1\\ &=\sum_{i=1}^n(2^{i+1}-2)-\sum_{i=1}^ni\\ &=2\sum_{i=1}^n2^{i}-\sum_{i=1}^n2-\sum_{i=1}^ni\\ &=2(2^{n+1}-2)-2n-\frac12n(n+1)\\ &=2^{n+2}-4-\frac52n-\frac12n^2\\ \end{align} $$


7

Stirling's approximation gives a pretty tight bound for the natural logarithm of $n!$ (but not an exact value), and then the base-10 logarithm is only a matter of dividing by $\ln 10$.


6

For good ballpark estimates, use the fact that $2^{10}=1024\approx 1000$. So $10$ doublings is about the same as multiplying by $1000$. Since $10^{82}=10\times 1000^{27}$, we can see that $270$ doublings will get us past $10^{81}$. There is some slack, and another $3$ doublings get us over.


5

Recall the geometric series (see Wikipedia): for any $y$ with $|y|<1$, $$\frac{1}{1-y}=1+y+y^2+\cdots=\sum_{k=0}^\infty y^k.$$ Therefore, for any such $y$, we also have $$\frac{y}{1-y}=y+y^2+y^3+\cdots=\sum_{k=1}^\infty y^k.$$ Now let $y=e^{-x}$ (though observe that we need $x>0$ to have $e^{-x}<1$).


5

They do not provide a derivation, but this is actually written up in Wikipedia. I use the standard notation $e(x) = \exp(2 \pi i x)$. Assuming $\gcd(k,n) = 1$, we have $$ \sum_{x \in \mathbb{Z}/n \mathbb{Z}} e\left(\frac{kx^2}{n} \right) = \left\{ \begin{array}{lcl} \varepsilon_n \left( \frac{k}{n} \right) \sqrt{n} & & n \equiv 1 \pmod{2}\\ 0 ...


5

Clearly, $$\begin{align*} \int_{0}^{\infty} \sin mx \sin nx \; dx &= \frac{1}{2}\int_{-\infty}^{\infty} \sin mx \sin nx \; dx \\ &= \frac{1}{4}\int_{-\infty}^{\infty} (\cos (m-n)x - \cos(m+n)x ) \; dx \\ &= \frac{1}{4}\int_{-\infty}^{\infty} (e^{i(m-n)x} - e^{i(m+n)x}) \; dx \\ &= \frac{\pi}{2}(\delta(m-n) - \delta(m+n)), \end{align*}$$ in ...


4

You have that $$0.3 + 0.03 + 0.003 + \cdots = \sum\limits_{n = 1}^\infty {\frac{3}{{{{10}^n}}}} $$ Now use that $$\sum\limits_{n = 1}^\infty {{a^n}} = \frac{a}{{1 - a}}$$ whenever $|a|<1$


4

If $x, y, z>1$, then $y^z, z^x, x^y \geq 2$ so $x^2y^2z^2 \mid x^{y^z}y^{z^x}z^{x^y}=5xyz$ so $xyz \mid 5$, impossible since $x, y, z \geq 2$. Thus at least one of $x, y, z$ is $1$. The equation is cyclic, so we may assume $z=1$, so the equation becomes $x^yy=5xy$ so $x^{y-1}=5$. It is now clear that we must have $x=5, y=2$. Therefore all solutions are ...


4

Unfortunately your intuition is incorrect, for kind of an annoying reason: you can find smaller sets of elements, of cardinalities $k$ and $m-k$, which individually sum to $0$. For example, take $u=30$ and $m=5$. Then $x_0+x_{15}=0$ and $x_1+x_{11}+x_{21}=0$, and so the set $\{x_0,x_1,x_{11},x_{15},x_{21}\}$ has the right sum but not the nice form you hoped ...


4

Write : $L^2=L(L-1)+L$ and use derivative. For $ L \geq 2$ : $$L^2 s^L = s^2L(L-1)s^{L-2}+ s Ls^{L-1}= s^2(s^L)'' + s (s^L)'$$ We get : $$\sum_{L=0}^M L^2s^L=0^2+1^2 s + s^2 \left(\sum_{L=2}^{M} s^L \right)''+s \left(\sum_{L=2}^{M} s^L \right)'$$


4

The $n$th step is $2^n$. If you want $2^n\geq A$, then you want $$n = \log_2(2^n) \geq \log_2(A) = \frac{\ln(A)}{\ln(2)} = \frac{\log_{10}A}{\log_{10}(2)}.$$ So the first $n$ at which $2^n\geq A$ will be the least positive integer greater than or equal to $\log_2(A)$, which is denoted $$\left\lceil \log_2A \right\rceil.$$


4

Here are two facts you might find useful: $$e^{2y+2} = e^2 e^{2y};$$ $$\frac{(2(y+1))^n}{n!} = \frac{2^n}{n!}(y+1)^n.$$


4

Well if $x,y\in \mathbb{R}$ then by definition \begin{equation}\exp(x)\exp(y)=(\sum_{k=0}^{\infty}\frac{x^k}{k!})(\sum_{k=0}^{\infty}\frac{y^k}{k!}) \end{equation} The Cauchy's Multiplication Theorem tells as that \begin{equation}\sum_{k=0}^{\infty}\sum_{k=0}^{n}a_kb_{n-k}=\sum_{k=0}^{\infty}a_k\sum_{k=0}^{\infty}b_k\end{equation} when at least one of the ...


3

Since we only have linear powers of $x$ (that is, powers of the form $ax+b,$ where $a,b$ constants with $a\ne 0$), then one approach we can take (that puts off logarithms until the very end) is this: $$7^x=5^{x-4}\\7^x=\frac{5^x}{5^4}\\7^x=\frac{5^x}{625}\\7^x\cdot625=5^x\\625=\frac{5^x}{7^x}\\625=\left(\frac57\right)^x$$ At this point, we can take a ...


3

Let $$ S^{(n)}(\lambda)=\sum_{k=0}^\infty\frac{\lambda^{nk}}{(k!)^n}. $$ Then $$ \lim_{n\to\infty}S^{(n)}(\lambda)=\begin{cases} 0 & \text{if }0\le\lambda<1,\\ 2 & \text{if }0\le\lambda=1,\\ \infty &\text{if }\lambda>1. \end{cases} $$ Let's prove it. If $0\le\lambda<1$ then $$ 1\le ...


3

Hint: Let $x = e^{5\delta}$. Notice that $e^{10\delta} = (e^{5\delta})^2$. Then we have $$316.45 = 100x^2 + 100x$$ Solve as usual for $x$, and retrieve $\delta$ afterwards.


3

Let's note that $$(2^1 - 1) + (2^2 - 1) + \cdots + (2^k - 1) = 2^{k+1} - 2-k$$ where we have used the geometric series. Thus, the desired sum is actually $$\sum_{k=1}^n{2^{k+1}-2-k}$$. As this is a finite sum, we can evaluate each of the terms separately. We get the sum is $$2\left(\frac{2^{n+1}-1}{2-1}-1\right) - 2n- \frac{n(n+1)}{2} = 2^{n+2}-4 - ...


3

Using the series definition of exponential: $$ e^{iG\lambda}A e^{-iG\lambda} = \sum_{p=0}^\infty\frac{(iG\lambda)^p}{p!}A\sum_{q=0}^\infty\frac{(-iG\lambda)^q}{q!} = \sum_{p=0}^\infty\sum_{q=0}^\infty(-)^q\frac{(i\lambda)^{p+q}}{p!q!}G^pAG^q=\\ \sum_{s=0}^\infty\sum_{d=0}^s(-)^d\frac{(i\lambda)^s}{d!(s-d)!}G^{s-d}AG^d=\\ ...


3

HINT: From the geometrical sequence, you know that $$ \sum_{n=0}^Nq^n =\frac{1-q^{N+1}}{1-q}, \quad q\neq 1. \tag1 $$ By differentiating with respect to $q$ $$ \sum_{n=0}^Nnq^{n-1} =\partial_q \left(\frac{1-q^{N+1}}{1-q}\right), \quad q\neq 1. \tag2 $$ Then apply it with $q=e^{2i\pi x}$, observing that $\displaystyle ...


3

This is just an application of the chain rule $$ \frac{\partial\mathrm{log}(\mathrm{exp}(w_1 * x_1 + b_1) + \mathrm{exp}(w_2 * x_2 + b_2))}{\partial w_1} = \frac{\partial}{\partial w_1}(log(f(w_1)) = \frac{\frac{\partial f(w_1)}{\partial w_1}}{f(w_1)} $$ So your final answer is: $$ \frac{x_1\mathrm{exp}(w_1x_1+b_1)}{\mathrm{exp}(w_1 * x_1 + b_1) + ...


3

This is a partial sum of a geometric series: $$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right| =\left| \frac{1}{n} \sum_{m=1}^n q^m \right| =\left| \frac{q}{n} \frac{1-q^n}{1-q}\right| \le \frac{2}{n\left|1-q\right|}$$ with $q:=e^{(2 \pi)^2 i h}$. If your question is whether there is an upper bound for this independent of $h$, the ...


3

Every positive integer $n$ is either even, i.e., of the form $n=2k$ for $k \geq 1$, or odd, i.e., of the form $n=2k-1$ for $k \geq 1$. More formally, if $f:\mathbb{N}\rightarrow\mathbb{R}$, then $$\sum_{n=1}^\infty f(n)=\sum_{\text{odd } n=1}^\infty f(n)+\sum_{\text{even } n=1}^\infty f(n)$$ since addition of real numbers is commutative and hence ...


3

Try to make the inner expression look like a derivative: $$ \begin{align} \sum_{L=0}^M\left(Ls^{L-1}\right)sL & =s\sum_{L=0}^M\left(\partial_ss^L\right)L\\ & =s\partial_s\sum_{L=0}^Ms^LL\\ & =s\partial_s\sum_{L=0}^M\left(Ls^{L-1}\right)s\\ & =s\partial_s\left(s\sum_{L=0}^M\left(Ls^{L-1}\right)\right)\\ & ...


3

Consider, instead, the series: $$\begin{align} \sum_{n=0}^\infty nx^n &= x\sum_{n=0}^\infty nx^{n-1} \\ &= x\sum_{n=0}^\infty\frac{d}{dx}x^n \\ &= x\frac{d}{dx}\sum_{n=0}^\infty x^n \\ &= x\frac{d}{dx}\left(\frac{1}{1-x}\right) \\ &= x\left(\frac{1}{(1-x)^2}\right) \\ &= \frac{x}{(1-x)^2} \end{align}$$ This converges whenever ...


3

Notice that: \begin{align*} 3^{x-5} + 3^{x-7} + 3^{x-9} &= 3^{4 + (x-9)} + 3^{2 + (x-9)} + 3^{x-9} \\ &= (3^4)3^{x-9} + (3^2)3^{x-9} + 3^{x-9} \\ &= (81)3^{x-9} + (9)3^{x-9} + (1)3^{x-9} \\ &= (81 + 9 + 1)3^{x-9} \\ &= (91)3^{x-9} \\ \end{align*}


3

Say you have a random variable $Y=aX$, where X is a random variable and $a$ is a scalar. Then $$M_Y(t)=M_{aX}(t)=E[e^{t(aX)}]=E[e^{(ta)X}]=E[e^{(at)X}]=M_X(at)$$ Since you're taking an average, then your new random variable is the sum of your $Y_i$'s, each multiplied by the scalar $\frac1{n}$. By properties of the moment generating function, ...


3

1) The moment generating function of a sum of independent random variables is the product of the individual moment generating functions. 2) If $W=aV$, where $a$ is a constant, then the moment generating functions $M_V(t)$ and $M_W(t)$ are related by the equation $$M_W(t)=M_V(at).$$ If your exponentials $Y_i$ have parameter $\lambda$, then each has moment ...


2

Hint: $$\sum_{m=n}^\infty a^nb^m=a^n\sum_{m=n}^\infty b^m = a^n\sum_{m=n}^\infty (b^n)(b^{m-n})=a^nb^n\sum_{m=n}^\infty b^{m-n}=a^nb^n\sum_{k=0}^\infty b^k.$$



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