# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $0 <a_n < 1$ ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 $z<1$, $$\sum_{k = 0}^\infty z^{2 k} = \frac{1}{1 - z^2}\ .$$
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### 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 ...
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### 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 ...