# Tagged Questions

37 views

### How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
70 views

### Mathematical Induction Problem with Fraction

$$(3n-2)^2=\frac{n(6n^2-3n-1)}{2}$$ I can't seem to solve it out to the point where I can prove it right or wrong. I always hit some sort of roadblock where I don't have enough info to prove it ...
31 views

### Recursive sequence problem

$$U(n+1) = (6+U(n))^{1/3},\text{ and } U(0) = 1.$$ Prove by induction that for all positive integers $n, U(n)$ is increasing. Prove by induction that for all positive integers $n, U(n) \leq 2$ ...
62 views

### Consider the sequence defined recursively by $U(n+1) =\frac{1}{3-U(n)}$ and $U(0) = 2$. [closed]

Prove by induction that for all positive integers $n, U(n)$ is decreasing Prove by induction that for all positive integers $n, U(n) > 0$ (namely, the sequence is bounded from below) Does the ...
78 views

### Proving that nth derivate of $x e^{-x}$ is $(-1)^n (e^{-x})(x-n)$ by induction.

I'm quite stuck on this. How would you prove that the $n^{th}$ derivative of $x e^{-x}$ if the $(-1)^n (e^{-x})(x-n)$ by induction? I did: $\frac{d}{dx}(x e^{-x})=(e^{-x}) - x(e^{-x})$ Now I have ...
40 views

### Proof by induction valid or not?

Prove by induction the following: $$\sum_{i=0}^n x^i = \frac{1-x^{n+1}}{1-x}$$ We want: $$x^0+x^1+ \ldots + x^n = \frac{1-x^{n+1}}{1-x}$$ I try this for $i=1$ and it works, so I have an initial ...
128 views

### Proof that $\binom{ n}{k} \in \mathbb N$

This problem is from Spivak. Give another proof that $\binom{n}{k}$ is a natural number by showing that $\binom{n}{k}$ is the number of sets of exactly $k$ integers each chosen from $1, \ldots,n$. I ...
114 views

312 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 ...
19 views

### Conjecture based on limited trail followed by inductive proof

My syllabus says: recognise situations where conjecture based on a limited trail followed by inductive proof is a useful strategy, and carry this out in simple casses e.g. find the nth derivative ...
89 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 ...
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: ...
47 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 ...
99 views

81 views

76 views

### Proof for power functions

Which is greater? $\sqrt{n}^{\sqrt{n+1}}$ or $\sqrt{n+1}^\sqrt{n}$ I know that $\sqrt{n}^{\sqrt{n+1}}$ is greater but I tried using induction and I couldn't figure it out. Thanks for the help.
170 views

### Proving ${(n+2)!}< \left(n(\sqrt{2}-1)+\sqrt{2}\right)^{n+2}$

While simplifying an inequality, this inequality was derived: $${(n+2)!}< \left(n(\sqrt{2}-1)+\sqrt{2}\right)^{n+2},\quad\quad\quad\quad n\in \mathbb{N}$$ Do you have any idea to prove it? It is ...
228 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 ...
### Induction and Integral Question (sum of 5th powers, integral of $x^5$)
I am quite lost on this question: (a) For $n\in \mathbb{N}$, use induction to show that $$\sum_{k=1}^{n}k^{5}=\frac{2n^{6}+6n^{5}+5n^4-n^2}{12}$$ (b) Fix $b>0$. Use the definition of ...