0
votes
2answers
37 views

Prove by induction that $(1+x)^n \geq 1+nx$ [duplicate]

Prove by induction that $\forall x \in \mathbb{R}, x \geq -1, \forall n \in \mathbb{N},n \geq 0$ that $$(1+x)^n \geq 1+nx$$ First of all I have a problem with x being a real number, how can I use ...
1
vote
0answers
44 views

Inductive proof about Jensen's inequality

The base case is easy. For the inductive step, i take $\lambda$ and $x$ to be as given, and then when I consider $f(\lambda_1 x_1 + . . . + \lambda_n x_n + \lambda_{n+1} x_{n+1})$ I get this is ...
0
votes
1answer
23 views

For $f: \mathbb{R}^n \to \mathbb{R}$ homogenous, show that $\sum_{i=1}^n x_i \frac{\partial f}{\partial x_i}(x_1, \dots ,x_n)= kf(x_1, \dots , x_n)$

Definition: A function $f: \mathbb{R}^n \to \mathbb{R}$ is said to be homogenous of degree $k$ if $\forall t \in \mathbb{R}$ and $(x_1, \dots , x_n) \in \mathbb{R}^n$ the equations $f(tx_1, \dots , ...
0
votes
2answers
44 views

Proofs for $g(x)=e^{-1/x^2}$ when $x\neq0$, and $g(x)=0$ when $x=0$

Sorry for the non-descriptive title - the question is a bit long. I have $g(x)$ as in the title, and we proved previously that $g'(0)=0$ using L'Hôpital's rule. Now I must show by induction that ...
1
vote
0answers
43 views

Proof by induction on contraction mapping?

Let $k:[0,1] \times [0,1] \to \mathbb{R}$ be continous, and $x(t) = \int_0^t k(t,s)x(s)ds$ for $0 \leq t \leq 1$. Not let $Tx(t) = \int_0^t k(t,s)x(s)ds$ and suppose $sup_{0 \leq t, s \leq 1}|k(t,s)|= ...
0
votes
0answers
46 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 ...
0
votes
2answers
31 views

Trying to prove by induction but do not know where to start (Analysis)

I understand how induction works but I am stuck on how I should approach this problem. I know I could start with the base case, but I'm not sure if my approach would be a solid proof. Here is the ...
2
votes
2answers
41 views

Got stuck in proving monotonic increasing of a recurrence sequence

I'm stuck with the prove of the following recurrence sequence which was part of and old exam. $a_1:=\frac{1}{2}, a_{n+1}:=a_n(2-a_n)$ for $n \in \mathbb{N}$ I have to show that $0 <a_n < 1$ ...
0
votes
0answers
34 views

Induction proof help

Let $x_1=1$ and $x_{n+1}=\frac{1}{x_n+1}$. Prove that for all $n \in \mathbb N$ $x_n >0$. So, $x_1=1>0$. Suppose $x_n>0$. I need to show that $x_{n+1}=\frac{1}{x_n+1}>0$. Isn't it ...
1
vote
1answer
32 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 ...
1
vote
2answers
56 views

Sum from $k=1$ to $n$ of $k^3$

$$\sum_{k=1}^n k^3 = \left(\frac{1}{2}n(n+1) \right)^2$$ I want to prove this using induction. I start with $(\frac{n}{2}(n+1))^2 + (n+1)^3$ and rewrite $(n+1)^3$ as $(n+1)(n+1)^2$, then factor out ...
1
vote
1answer
50 views

Integration using induction question

Assume $f : [0, 1] \to \mathbb{R}$ is continuous and arbitrarily often differentiable on $(0, 1)$ (i.e. $f$ is smooth). Denote by $f^{m}$ the $m\text{-th}$ derivative of $f$ with $m∈\mathbb{N}$ and ...
1
vote
1answer
53 views

Am I understanding induction correctly?

Here is an induction proof that I have written for my homework and I want to know if I am understanding this correctly: Prove that for: $ \sum\limits_{i=0}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ My proof: ...
0
votes
2answers
44 views

Bernoulli's inequality by induction

I'm proving Bernoulli's inequality by induction but I noticed something strange. See wikipedia proof: http://en.wikipedia.org/wiki/Bernoulli's_inequality Notice how they multiply both sides of the ...
2
votes
3answers
68 views

Prove $2^n\cdot n! ≤ (n+1)^n$ by induction.

An induction I'm struggling with. Prove $2^n\cdot n! ≤ (n+1)^n$ by induction. An idea was to show that $2^n\cdot n! ≤ 1+n^2$ since $1+n^2 ≤ (n+1)^n$ using Bernoulli. However the inequality is ...
0
votes
1answer
46 views

How to prove $x_{n+1}y_n-x_ny_{n+1}=2^{3n}\sqrt{7}$

For $n \in \mathbb{N}$ let $z_n=(1-i\sqrt {7})^n, ~x_n = Re~z_n,~y_n = Im~z_n$. I want to show the following: $x_{n+1}y_n-x_ny_{n+1}=2^{3n}\sqrt{7} ~~~(n \in \mathbb{N})$ My only idea was to show ...
1
vote
2answers
58 views

Show by induction $|a_1-a_2+a_3-\ldots \pm a_n| \leq |a_1|$

The assumptions are that $(a_n)$ is a decreasing sequence with $(a_n) \to 0 $, that is all terms are nonnegative. It is easy to see that the subtracted terms are always at least as great as the added ...
0
votes
1answer
50 views

Proof By Induction Quest.

Prove that for all integers $n$ greater then $1$: $$ 1+\frac {1}{4} +\frac {1}{9} +\cdots+\frac {1}{n^2} <2-\frac{1}{n} $$ First verify that $p(n)$ is true for $n = 2$ $\frac { 1 }{ 4 } ...
0
votes
2answers
153 views

Use product rule and mathematical induction to show that $f^n$ is differentiable on $I$

Suppose that $f$ is differentiable on $I$. Use the product rule and mathematical induction to show that $f^n$ (the function f is raised to the nth power) is differentiable on $I$ for every positive ...
-1
votes
1answer
87 views

Prove that $f(mx)=mf(x)$ for all $x\in\mathbb{R} $ and $m\in\mathbb{Z} $ [closed]

Suppose that $f \colon\mathbb{R}\to \mathbb{R}$ satisfies the functional equation $f(u+v) = f(u)+f(v)$ for all $u,v \in\mathbb{R} $. Prove that $f(mx)=mf(x)$ for all $x\in\mathbb{R} $ and ...
3
votes
2answers
72 views

Prove that $ \left(1-\frac{1}{n}\right)^n > \frac{1}{6} $ for $n\geq 2$

Prove that $ \left(1-\frac{1}{n}\right)^n > \frac{1}{6} $ for $n \in \mathbb{N}$, $ n\ge 2$ Indeed, the affirmation is true even if $n$ is not a natural ($ n\geq 2 $ ) and we can prove it using ...
0
votes
1answer
16 views

Induction on a triangular equality related problem

I have to prove $|a_1,a_2,\dots,a_n|\leq |a_1|+|a_2|+\dots+|a_n|$ For $n$ numbers $a_1,a_2,\dots,a_n,\dots$. How can I start this problem? I used this so far (triangular equality set up or rather an ...
0
votes
4answers
102 views

Math analysis $n^2>n+1$ [closed]

So by induction $n^2<n+1$ is $Pn$ and holds for all integers n less than and equal to 2 For $P_n+1$ $(n+1)+1<(n+1)^2 \\ <n^2+2n+1 \\<(n+1)^2+2(n+1)+1$ Is the the correct way to ...
4
votes
5answers
342 views

Prove that $1^3 + 2^3 + \cdots + n^3 < n^4$.

I am trying to prove the following: $1^3 + 2^3 + \cdots + n^3 < n^4$ if $n \in \mathbb{N}, n>1$ by induction. From there, I am to prove that the sum is $< \frac{n^4}{2}$ if $n>2$. My ...
1
vote
3answers
330 views

Showing Whether a Sequence is Bounded Above or Not

I am trying to solve the following problem about a sequence: Consider the sequence ${a_n}$ where $a_n = 1 + \frac{1}{1 \cdot 3} + \frac {1}{1 \cdot 3 \cdot 5} + \frac {1}{1 \cdot 3 \cdot 5 \cdot 7} + ...
1
vote
2answers
165 views

Why couldn't use the mathematical induction?

We can use mathematical induction which is deduced from Peano axioms and illustrated on Terence Tao's Real Analysis(here it is) Axiom 2.1 $0$ is a natural number. Axiom 2.2 If $n$ is a ...
2
votes
1answer
71 views

show by induction if there exists a $n_0 \in \mathbb N $such that $n\geq n_0 , n! \gt 2n^3$

I tried and I got there doesn't exist such a $n_0$ However, I dun think I have a formal proof for this. My approach is, First assume there is such a $n_o$ exist and start my calculation with ...
0
votes
1answer
145 views

Prove by Induction using Baseline and splitting into LHS & RHS?

I'm having trouble with this equation mainly because it has a couple of odd things with it, and its these that have thrown me off as i'm not to sure how to tackle them. The equation is: ...
0
votes
2answers
120 views

Induction proof

Let $0<a_1<b_1$ and define, for $n\in \mathbb{N}$, $$a_{n+1}=(a_nb_n)^{1/2}, \ b_{n+1}=\frac{1}{2}(a_n+b_n).$$ by induction show that $a_n<b_n$. Show that $a_n$ and $b_n$ converge ...
0
votes
1answer
68 views

Show by induction -Analysis

Show by induction that for all $z\notin\{-1,1\}$ one has $$\sum_{k = 0}^n z^{2 k} = \frac{1-z^{2n+2}}{1-z^2}\ .$$ Deduce that if $z<1$, $$\sum_{k = 0}^\infty z^{2 k} = \frac{1}{1 - z^2}\ .$$
3
votes
1answer
168 views

How does one determine which variables to do induction on?

I have been struggling with this all day. When one does mathematical induction, how does one choose when to induct with one variable, or with more than one? I have been working through Tao's ...
2
votes
1answer
215 views

How do I implement an induction argument to show that we may assume that a partition $Q$ of $[a,b]\in{R}$ which refines $P$ has only one more point?

There is a lemma (32.2) in Kenneth A. Ross's Elementary Analysis text, that states: Let $f$ be a bounded function on $[a,b]$. If $P$ and $Q$ are partitions of $[a,b]$ and $P\subseteq{Q}$, then ...
1
vote
3answers
80 views

Induction prove, how to come to $n\cdot(n+1)$

I am trying to solve an induction problem. Here are the steps for the example. Prove this equation $$ 1\cdot2 + 2\cdot3 + 3\cdot 4 + 4\cdot 5+\dots + \cdots +(n-1)\cdot n ...
1
vote
2answers
79 views

prove, that following formula is correct

As in the statement, I got problems with:$$\binom{n}{0} +\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2[\frac{n}{2}]}=2^{n-1}$$ I started with Newton conjecture, trying to work with ...
2
votes
2answers
143 views

simple induction, why does it go this way?

I have a simple question involving induction. $$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\geq \frac{1}{2}.$$ I think that this does not go with most common version of induction, because of the ...
14
votes
5answers
2k views

Why doesn't induction extend to infinity? (re: Fourier series)

While reading some things about analytic functions earlier tonight it came to my attention that Fourier series are not necessarily analytic. I used to think one could prove that they are analytic ...