## Prove that the sequence $a_1, a, a_2, a, a_3, a,\ldots$ converges iff $a_1,a_2,a_3,\ldots$ converges November 28

Prove that the sequence $a_1, g, a_2, g, a_3, g,\ldots$ converges to $g$ iff $a_1,a_2,a_3,\ldots$ converges to $g$.Obviously, if $a_1, g, a_2, g, a_3, g,\ldots$ converges to $g$, then its subsequence $a_1,a_2,a_3,\ldots$ converges to $g$. On the cont

## Surface Integral problem. May be some miss conception. November 28

Evaluate $\int \int \vec A.\hat n dS$, where$\vec A = 18z\hat i - 12\hat j + 3y\hat k$ and $S$ is that part of the plane $S$ $2x+3y+6z = 12$ which is located in the first octant. The surface ...

## Floor and Ceiling functions November 28

I have been trying to proof ⌊log_2(⌈n/k⌉)⌋ = ⌊log_2(n/k)⌋, but I never learned any rules with floor and ceiling functions. I am not sure if this theorem is true either. So my question is: Is it safe to say ⌊log_2(⌈n/k⌉)⌋ = ⌊log_2(n/k)⌋ ?

## Prove that for every integer $n\geq 0$, $1+3n\leq 4^n$. November 28    2

Prove that for every integer $n\geq 0$, $1+3n\leq 4^n$. Proof:Let the property $P(n)$ be the inequality $$1+3n\leq 4^n.$$ Establishing $P(0)$, we see that $1+3(0)=1$ and $4^0=1$, hence $P(0)$ is true. Suppose $k$ is any integer with $k\geq 0$ such th

## For every integer $n \geq 1$, prove that $3^n \geq n^2$. November 28    3

It's been a while since I've done induction, and I feel like I'm missing something really simple. What I have is this:Base Case: $n=1$ $$3^n \geq n^2 \implies 3 \geq 1$$Inductive Hypothesis For all integers $1 \leq n < n+1$: $$3^n \ge n^2$$Inductive

## Using Cauchy's Integral Formula to show that $f(z) = e^z$ for every $z$ with $z \lt 1$ November 28    1

I'm learning about complex analysis and need some help with this problem :Given $f : \Bbb C \rightarrow \Bbb C$ analytic with $f(z) = e^z$ for every $z$ with $|z| = 1$. Show that $f(z) = e^z$ for every $z$ with $|z| \lt 1$. (Hint : use Cauchy's Inte

## Set Sum Partition problemPigeon hole Application November 28

Prove that from every set of 2n integers, you can chose a subset of n elements, such that the sum is divisible by n.

## If $f(z)$ is entire and $f(z) \le \log(2+z)$ for every $z \in \Bbb C$ show that $f$ is constant November 28

I'm learning about complex analysis and need to verify my work to this problem since my textbook does not provide any solution:If $f(z)$ is entire and $|f(z)| \le \log(2+|z|)$ for every $z \in \Bbb C$ show that $f$ is constant. Here's my attempt : Su

## Integrate $\int \frac{\arctan\sqrt{\frac{x}{2}}dx}{\sqrt{x+2}}$ November 28

$$\int \frac{\arctan\sqrt{\frac{x}{2}}dx}{\sqrt{x+2}}$$ I've tried substituting $x=2\tan^2y$, and I've got: $$\frac{1}{\sqrt2}\int\frac{y\sin y}{cos^4 y}dy$$ But I'm not entirely sure this is a good thing as I've been unable to proceed any further fr

## Limit of zeta function in $x = 1$ November 28    1

How can I prove that $\lim_{x \rightarrow 1}{\sum_{n=1}^{\infty}{\frac{1}{n^x}}} = \infty$? My idea is to show that we can exchange the positions of limit and sum, obtaining the harmonic sum, that we know that diverges, but, are we able to do such ex

## Asymptotic expansion of $\sum_{n = 2}^{x} \dfrac{1}{\log(n)}$ and $\sum_{n=1}^{x}\dfrac{1}{\sum_{k=1}^{n}k^{-1}}$ November 28    1

Presumably \begin{align} \operatorname{Li}(x) = & \sum_{n = 2}^{x} \dfrac{1}{\log(n)}+ O(\log(x))\\ \end{align} where \begin{align} \operatorname{Li}(x) = & \int_{2}^{x}\dfrac{1}{\log(t)}\operatorname{d}t \end{align} By Euler-Maclaurin approximati

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