2
votes
2answers
36 views

Proof by Induction: $(1+x)^n \le 1+(2^n-1)x$

I have to prove the following by induction: $$(1+x)^n \le 1+(2^n-1)x$$ for $n \ge 1$ and $0 \le x \le 1$. I start by showing that it's true for $n=1$ and assume it is true for one $n$. ...
0
votes
1answer
32 views

Complex Polynomial That is n Times Differentiable: A Concern

I'm looking at a question that asks me to show that: If a function $f$ is known to be $n$-times differentiable in a domain $D$ and if $\forall{z\in{D}}\ \ f^{(n)}(z)=0$, then $f$ is a polynomial ...
2
votes
1answer
39 views

sum of an arctan series using mathematical induction

How to solve this problem using mathematical induction: $$\arctan (1) + \arctan \Big(\frac13\Big) + ... + \arctan \bigg(\frac{1}{n^2+n+1}\bigg)=\arctan (n+1)$$
0
votes
0answers
48 views

Taylor series like polynomials

Let $U$ be an open subset of $R^n$ and $f:U\rightarrow \mathbb{R}$ a function and $x\in U$ such that in a small neighbourhood of $x$ and for $\epsilon \in \mathbb{R^b}$ sufficiently small we have the ...
1
vote
1answer
35 views

Small question about inductive proof about rational sequences

I am writing an inductive proof about this: the description is not terribly important so you don't have to read that. here's my question: let $P(n)$ be the statement that $x_n$ is a rational ...
2
votes
6answers
314 views

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\ldots+a_n<\frac{1}{2}$, prove that $(1+a_1)(1+a_2)\ldots(1+a_n)<2$.

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\cdots+a_n<\frac{1}{2}$, prove that $$(1+a_1)(1+a_2)\cdots(1+a_n)<2$$ I've tried Hölder's inequality (the same result can easily be derived using ...
1
vote
2answers
98 views

Proof of an inequality by induction[solved]

So I have this inequality and I just can't figure out how to prove it: Prove that ($\forall n\in \mathbb N)$ $$\sum_{k=1}^n \frac{1}{(k+1)\sqrt k}<2.$$ I've figured that for $n=1$ the inequality ...
1
vote
2answers
47 views

An induction problem.

I am trying to prove the following problem by induction on $n$. Let $T: (0,1]\rightarrow (0,1]$ be given by $T(x)=\left\{ \begin{array}{ll} 2x & \quad \text{if} \hspace{4mm} ...
3
votes
4answers
85 views

proof by induction : $n^n \ge 2^{n-1} n!$

I am trying to show that $$\begin{matrix} n^n \ge 2^{n-1} n! & \text{(1)} \end{matrix}$$ I tried to solve it for n=n+1 $$(n+1)^{n+1}=(n+1)^n(n+1) \ge n^n(n+1) \ge 2^{n-1}n!(n+1)= ...
1
vote
1answer
651 views

Proof for Strong Induction Principle

I am currently studying analysis and I came across the following exercise. Proposotion 2.2.14 Let $m_0$ be a natural number and let $P(m)$ be a property pertaining to an arbitrary natural ...
0
votes
1answer
47 views

Check Logic for Induction Of T(n) Equation

I've just finished an induction equation, however i'm a little hit and miss about whether its actually right on not. Mainly in my working out, I'd really appreciate if you could have a look at it and ...
7
votes
4answers
421 views

Given that $f(1)= 2013,$ find the value of $f(2013)$?

Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$. Given that $f(1)= 2013$, find the value ...
0
votes
1answer
95 views

Why this inequality yields at most exponential growth?

Let $\Omega=\mathbf{R}^{n-j}\times\omega$, where $\omega\subset\mathbf{R}^j$ is a smooth bounded domain. Consider a function $u:\overline\Omega\rightarrow\mathbf{R}$ that satisfies $$u(x,y)+k\leq ...
0
votes
4answers
471 views

Using induction, prove that $\sum_{k=1}^n\frac 1 {(2k-1)(2k+1)}= \frac 12-\frac 1 {4n+2}$ when $n$ is in $\mathbb{N}$

I can prove the base case and get that $1/3=1/3$, but i can't get any further with the $n+1$ case. Can someone help me? I was told to conjecture a formula for the sum $$\frac{1}{3} + \frac{1}{{3 ...
1
vote
1answer
159 views

Prove that if $n \in \mathbb{N}$, $n\ge 1$

As the title says. I encounter this problem in Bernd Schroeder's book of "Mathematical Analysis: A Concise Introduction", p.15. It essentially characterizes natural number from the axioms regarding ...
1
vote
1answer
164 views

Prove $2^n > n^3$ [duplicate]

Let $P(n)$ be the property: $2^n > n^3$. Let's use mathematical induction to prove that $P(n)$ is true for $n\geq10$. Basis: $P(10): 2^{10} > 10^3 \Leftrightarrow 1024 > 1000$ which is true. ...
3
votes
2answers
277 views

Is this analysis problem involving induction flawed?

I was recently asked to help someone with the following question on their first year analysis course. Recall that $\mathbb{N}$ is the set of all positive integers. Use the principle of induction to ...