Questions about exponentiation

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7
votes
3answers
178 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me. It ...
3
votes
3answers
62 views

$x^y < y^x$ for $y\ll x$?

Sorry if this is a naive question; I am not very good at mathematics. It seems obvious that for many $x$ and $y$, $x^y < y^x$ if $y \ll x$, e.g. $2^{10} > 10^2$. If $x$ and $y$ are very close ...
1
vote
1answer
72 views

combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to calculate ...
0
votes
1answer
42 views

Why, or why not, is $5^{log_3(n)} \in \mathcal{O}(n^2)$?

Why, or why not, is $5^{\log_3(n)} \in \mathcal{O}(n^2)$ ? I tried transforming the logarithm to a base of 5, so that the logarithm and power cancel each other out. However, when I try to so I get ...
1
vote
1answer
61 views

Square root of $\frac{2}{2^x}$; how do I find $x$?

I have this: $$\sqrt{\frac{2}{2^x}} = 9.313225746154785 \times 10^{−10}$$ (sqrt(2/(2^x))) How should I find $x$? I know it's 61 for this case, but I'd like to know how to solve it for when I don't. ...
3
votes
1answer
35 views

Calculating limit in parts. Why possible?

Let $f$, continuous function, differentiable at $x=1$ and $f(1)>0$. Consider the following equation: $$\lim \limits_{x\to 1} ...
2
votes
4answers
113 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
1
vote
1answer
37 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
2
votes
2answers
84 views

Is $\lim_{x\to 0+} \frac{\ln(x)}{\ln(x)} = \frac{-\infty}{-\infty} = 1$

$$\lim_{x\to 0^+} \sin(x)^\frac{1}{\ln(x)} = ... = \exp \left(\lim_{x\to 0^+} \frac{\ln(\frac{\sin x}{x}) + \ln(x)}{\ln(x)}\right)$$ Now, from continuity we can evaluate each term separately. ...
3
votes
2answers
47 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
8
votes
5answers
907 views

Solving base e equation

So I ran into some confusion while doing this problem, and I won't bore you with the details, but it comes down to trying to solve $e^x - e^{-x} = 0$. I know to solve it, we can rewrite it as $e^x - ...
1
vote
0answers
46 views

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ primes. What are the first values of $U(n)$?

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ prime numbers (except for the first prime number: $2$). What are the first values of $U(n)$ up to ...
0
votes
1answer
51 views

How do they integrate this exponential?

Below, I tired to integrate te^(-j2pi*t) from 0 to 1. But am not getting what my professor got for n not equal to zero, which is also shown. I tried LIATE but am always getting something with an ...
2
votes
4answers
110 views

$(-1)^{0.2}=0.8090 + 0.5878i$ how can this be?

I'm working on a numerical analysis project (working with matlab a lot) and I noticed that when I ask for matlab to compute the exponent of a negative number, it gives wrong output when the exponent ...
4
votes
2answers
148 views

Combination of quadratic and arithmetic series

Problem: Calculate $\dfrac{1^2+2^2+3^2+4^2+\cdots+23333330^2}{1+2+3+4+\cdots+23333330}$. Attempt: I know the denominator is arithmetic series and equals ...
2
votes
2answers
24 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
0
votes
0answers
56 views

Exponential integral with $x^2$ and $\cos x$

The first part is just a Gaussian integral and the second is the modified Bessel function of the first kind for $n=0$, but I couldn't find any information and what to do with their summation. Any tips ...
0
votes
2answers
16 views

Find numbers who compose the sum of sequence numbers(nth root)

I have an sequence of numbers 1, 2, 4, 8, 16, 32 and so on. Given an number, which must be the sum of the sequence, for example: 44(which is the sum of 4+8+32). How do I know the number 44 is composed ...
0
votes
1answer
21 views

Algebraic Proof of Sum of Exponential Powers is Product of Exponentials

Can somebody provide a proof of the summation of powers law for the product of two exponentials, using only algebra and the Taylor series, no derivatives or calculus tricks?
2
votes
3answers
89 views

Solve exponential equation $3^x= 2^x+2$

How do we solve this? I can't think of an easy way.. Is there any way to solve it without using newton's method or other approximations? $3^x=2^x+2$
1
vote
1answer
49 views

Why do we define the complex exponential as we do?

Why do we define the complex exponential as we do? Defining $e^{z}$ as $e^{x}e^{iy}$ certainly seems to make sense, but I'm not sure the formal reason as to why it's defined like this. Was it from ...
7
votes
4answers
2k views

Non-integer powers of negative numbers

Roots behave strangely over complex numbers. Given this, how do non-integer powers behave over negative numbers? More specifically: Can we define fractional powers such as $(-2)^{-1.5}$? Can we ...
7
votes
4answers
272 views

The binomial formula and the value of 0^0

Here is the text from Knuth's The Art of computer programming, 1.2.6 F formula 14: Knuth doesn't give the proof of the statement. So, I tried to write it myself. To make binomial formula equal to ...
0
votes
0answers
21 views

Simplification of a power weighted alternating binomial sum

Given positive integers $T$, $n$ and $m$ and real number $p$ with $0< p < 1$, how can I simplify the following binomial sum: $$ \sum_{k=m}^{\lfloor ...
1
vote
0answers
14 views

Fast way to find exponential of a matrix dot product where one of them is diagonal

Suppose $Q$ is a dot product of diagonal matrix A and matrix B: $$ Q=A\cdot B= \left( \begin{matrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ ...
1
vote
4answers
72 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
8
votes
2answers
69 views

If any integer to the power of $x$ is integer, must $x$ be integer?

My apologies if this has been asked already, I've searched but couldn't find it... Let $x$ such that for every $y \in N$, $y^x$ is an integer. Does that necessarily mean that $x$ is an integer?
2
votes
1answer
177 views

Solving Weird Exponential Equations

I am working on my math homework when I encountered a difficult problem. I simplified the equation and substituted smaller numbers to get this: $n*2^n>10$ I have tried standard algebraic ...
0
votes
1answer
38 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
1
vote
3answers
41 views

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$ My try: $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$ $log_e{(x-2y)^2} = log_e{xy}$ ...
0
votes
1answer
48 views

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $
0
votes
5answers
47 views

Fraction with negative exponent fraction.

Q: $$\left(\frac{27 a^6 b^{-3}}{c^{-2}}\right)^{-2/3}$$ A: $$\frac{b^2}{9 a^4 c^{4/3}}$$ How in the world are they getting that?
4
votes
5answers
70 views

What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?

I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem. $$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$ I'm sure ...
1
vote
1answer
45 views

$\mathrm{Ei}(x)$, the exponential function, some question.

I have a question involving with $\mathrm{Ei}(x)$, define as $\int_{-x}^{\infty}e^u \cdot u^{-1} \mathrm{d}u$. My question is, when I have a expression say $\exp(x) \cdot \mathrm{Ei}(x)+1$. I want ...
1
vote
3answers
54 views

negative powers $(x^{-2} = 1/x^2)$

I need clarification for negative power of a number. I understand $x$ to the power of $2$ is equal to $x\cdot x$ But how $x$ to the power of $-2$ is equal to $\dfrac{1}{x^2}$ ?
1
vote
2answers
49 views

Definition of $a^b$ for complex numbers

Problem statement Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$ $(i)$ Verify that if $b \in ...
0
votes
1answer
61 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
10
votes
4answers
680 views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
1
vote
1answer
52 views

How are the cardinalities of the object images of adjoint functors related?

Here is a very silly question: Adjoint functors satisfy $$\mathrm{hom}_{\mathcal{C}}(FA,B) \cong \mathrm{hom}_{\mathcal{D}}(A,GB).$$ I consider numbers $a,b$ and read this as ...
4
votes
2answers
107 views

Limit of $x^x$ as $x$ tends to $0$

I am trying to solve the following limit: $$\lim \limits_{x\to0} x^x$$ The only thing that comes to mind is to write $x^x$ as $e^{x\ln{x}}$ and getting the right sided limit would be easy but I ...
0
votes
1answer
43 views

Something that grows faster than NP class of problem does

I have a theoretical question. F.ex. we have a NP-class of problem, i.e. which do need exponential time on deterministic Turing machine. Is there anything that is growing faster than exponent does. ...
0
votes
5answers
72 views

Factorial of zero is 1. Why? [duplicate]

Why is the factorial of zero, one. What is the mathematical proof behind it?
1
vote
1answer
71 views

Proof that irrational exponents exist, and are unique?

Can someone provide a proof that for any given irrational number, $b$, exponentiation by that number defined as a limit of rational powers always converges, and that if we choose a particular base, ...
-1
votes
1answer
28 views

Special cases of the $A^k$ matrix power

Let $A$ be az $n \times n$ real matrix. Suppose, that there is a fixed $k_0 \in \mathbb{N}_+$, and for this $k_0$ the matrix power $A^{k_0}$ has a closed formula, for example $A^{k_0}=k_0 A$ or ...
1
vote
0answers
53 views

If $f$ is holomorphic, then there is a holomorphic function $h$ such that $e^{h(z)}=f(z)$

Let $f:G\to\mathbb{C}$ denote a holomorphic function over a star-shaped domain $G$ and $f\ne 0$ on $G$. I want to show that it holds $\frac{f'}{f}$ is holomorphic There is a holomorphic function ...
0
votes
0answers
14 views

Finding the exponent in this situation

Basically how do I find an exponent of 2 that when multiplied with another number would bring the result closest to the positive side 1? Like this: $y = x \cdot 2^a$, where $y≥1$ has to be as small as ...
3
votes
3answers
46 views

$M$ matrix, $\mathrm{rank}\ M=1$. Prove that $det(e^M)=1$ iff $M$ is not diagonalizable

M is a $n\times n$ matrix over $\mathbb R$. with $\mathrm{rank}\ M=1$. Prove that $det(e^M)=1$ if and only if $M$ is not diagonalizable. I really don't know how to start thinking about this.. :/ I'd ...
1
vote
1answer
46 views

Golden-Thompson inequality and Lieb's theorem

On the [Wikipedia article][1] on "matrix exponential", they draw a relation between the Golden-Thompson inequality and Lieb's theorem. My questions are: It mentions that Lieb's thoerem "accomplishes ...
1
vote
1answer
17 views

Proving increasing function, base < 1, exponent increasing

For a fair lottery game where the odds of $1$ ticket winning are $1$ in $p$, where you can spend a total of $K$ dollars, and where you will spread your ticket purchases equally among $n$ draws, prove ...
0
votes
1answer
20 views

Question about calculating exponent of polynomial

$V=R_{3}[X] $ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$ I need to find $e^{T(1+x+x^{2}-x^{3})}$ I don't understand how to do it? what does it mean to calculate exponent ...